HOMEWORK

HOMEWORK: (click here for comments / solutions to the HW)

Please spend at least 1-2 hours a night reading the material/looking at the proofs/making sure you understand the details. Below is a tentative reading list and homework assignments. It is subject to changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

Week 12: May 5 - May 9:

Read through Chapter 21
Videos: Tues: https://youtu.be/xuyA_NTUCu0 Thurs: https://youtu.be/d4EpxCzkubo
Due Thursday, May 8: Assume two people randomly move in tic - tac - toe; what is the probability each wins (or that there is a draw)? Do this both through simulation and theoretically. The number of possible games is small enough (9! is about 362, 880 ) that we can just do the theory by brute force computation. If we had a bigger board it would be very different and then simulations would be the way to go.
Due May 15, 2025. \#1: Let $X, X_1, \dots, X_N$ be independent exponential random variables with parameter $\lambda$. Find the moment generating function for $X_i$. Directly using the moment generating function, prove the central limit theorem for $X_1+\cdots+X_N$ (i.e., mimic what we did for the Poisson). \#2: Write a question about probability and provide the answer, with a complete, correct argument. This \emph{must} be done in LaTeX. It does not matter how easy or hard your problem is -- so long as your argument is correct you will receive full credit. The purpose of this problem is to give you practice in making up problems. \#3: Write a question about probability that you would like to know the answer to -- you do not have to solve it (but you must LaTeX the problem). \emph{When submitting your homework this week, hand in all three problems together, and email the TeX file to me. Your filename should be yourlastname\underline{\ \ }yourfirstname\underline{\ \ }math341\underline{\ \ }hw11.tex.} \\

Week 11: April 28 - May 2:
- Read through Chapter 21
- Videos: Tues: No Class Thurs: https://youtu.be/fymK9-2jfrA
- Due Thursday, May 8: Assume two people randomly move in tic - tac - toe; what is the probability each wins (or that there is a draw)? Do this both through simulation and theoretically. The number of possible games is small enough (9! is about 362, 880 ) that we can just do the theory by brute force computation. If we had a bigger board it would be very different and then simulations would be the way to go.
- Due May 15, 2025. \#1: Let $X, X_1, \dots, X_N$ be independent exponential random variables with parameter $\lambda$. Find the moment generating function for $X_i$. Directly using the moment generating function, prove the central limit theorem for $X_1+\cdots+X_N$ (i.e., mimic what we did for the Poisson). \#2: Write a question about probability and provide the answer, with a complete, correct argument. This \emph{must} be done in LaTeX. It does not matter how easy or hard your problem is -- so long as your argument is correct you will receive full credit. The purpose of this problem is to give you practice in making up problems. \#3: Write a question about probability that you would like to know the answer to -- you do not have to solve it (but you must LaTeX the problem). \emph{When submitting your homework this week, hand in all three problems together, and email the TeX file to me. Your filename should be yourlastname\underline{\ \ }yourfirstname\underline{\ \ }math341\underline{\ \ }hw11.tex.} \\

Week 10: April 21 - 25:
- Read: Review Chapters 19 and 20, read Chapter 21
- Videos: Tues: https://youtu.be/PAgTajBx5ak Thurs: https://youtu.be/x-RNxpyi_4I
- Homework: Due Thursday April 24: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.
Week 9: April 14 - 18
- Read: Up to and including Chapter 20
- Video: Tues: https://youtu.be/0IqoNdzPI84 Thurs: https://youtu.be/e17FtuD-qJk
- Homework: Thursday, April 17: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk: https://www.youtube.com/watch?v=JoGFR4cDRu0. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
- Homework: Thursday, April 24: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.

Week 8: April 7 - 11
- Read: Chapters 15 to 18
- Video: Tues https://youtu.be/FjfRqQnJlLo Thurs: https://youtu.be/EZq9XKLSzPA
- Homework: Thursday, April 17: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk: https://www.youtube.com/watch?v=JoGFR4cDRu0. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 7: March 17 - 21
- Read: Chapters 12 to 15
- Video: Tues: https://youtu.be/E6o3AUEbGxM Thurs: https://youtu.be/BRTWawmZJLs
- Homework: Due Thursday April 17: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk: https://www.youtube.com/watch?v=JoGFR4cDRu0. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 6: March 10 - 14
- Videos: Tues: https://youtu.be/iXmE8jvOYAQ Thurs: https://youtu.be/mGQb4ZTTu0c
- Read: Make sure you have read through Chapter 11 (though I doubt we'll get through all of Chapter 11, have a strategic reserve of reading!)
- HW: Due Thursday March 27: #1: We toss $n$ fair coins. Every coin that lands on heads is tossed again. What is the probability density function for the number of heads after the second set of tosses (i.e., after we have retossed all the coins that landed on heads)? If you want, imagine you have left the room and return after all the tossing is done; what is the pdf for the number of heads you see? #2: Is there a $C$ such that $f(x) = C Exp(-x - Exp(-x))$ is a probability density function? Here $-\infty < x < \infty$. #3: Let $X$ be a discrete random variable. Prove or disprove: $E[1/X] = 1/E[X]$. #4: Let $X_1, \dots, X_n$ be independent, identically distributed random variables that have zero probability of taking on a non-positive value. Prove $E[(X_1 + \cdots + X_m) / (X_1 + \cdots + X_n)] = m/n$ for $1 \le m \le n$. Does this result seem surprising? Write a computer program to investigate when the random variables are drawn from a uniform distribution on [0,1]. #5: Let $X$ and $Y$ be two continuous random variables with densities $f_X$ and $f_Y$. (a) For what $c$ is $c f_X(x) + (1-c) f_Y(x)$ a density? (b) Can there be a continuous random variable with pdf equal to $f_X(x) f_Y(x)$? #6: The standard normal has density $\frac1{\sqrt{2\pi}} Exp(-x^2/2)$ (this means this integrates to 1). Find the first four moments. #7: Find the errorS in the following code:
- hoops[p_, q_, need_, num_] := Module[{},
  birdwin = 0;
  For[n = 1, n <= num,
  {
      If[Mod[n, num/10] == 0, Print["We have done ", 100. n/num, "%."]];
      birdbasket = 0;
      magicbasket = 0;
      While[birdbasket < need || magicbasket < need,
          {
              If[Random[] <= p, birdbasket = birdbasket + 1];
              If[Random[] <= q, magicbasket = magicbasket + 1];
          }]; (* end of while loop *)
      If[birdbasket == need, birdwin == birdwin + 1];
  }]; (* end of for loop *)
  Print["Bird wins ", 100. birdwin/num, "%."];
  Print["Magic wins ", 100. - 100. birdwin/num, "%."];
  ];
  
  hoops[.32, .33, 5, 100]

Week 5: March 3 - 7
- Read: Chapters 7 to 10.
- Video: Tues: https://youtu.be/N5jLFMOyW8A Thurs: https://youtu.be/gXubfwC0vJw
- HW \#4: Due Thursday, March 13: Note Mathematical Induction might be useful for some of these problems. \#1: Let $\{A_n\}_{n=1}^\infty$ be a countable sequence of events such that for each $n$, ${\rm Prob}(A_n) = 1$. Prove the probability of the intersection of all the $A_n$'s is 1. \#2: Prove the number of ways to match $2n$ people into $n$ pairs of 2 is $(2n-1)!!$ (recall the double factorial is the product of every other integer, continuing down to 2 or 1). \#3: Assume $0 < {\rm Prob}(X), {\rm Prob}(Y) < 1$ and $X$ and $Y$ are independent. Are $X^c$ and $Y^c$ independent? (Note $X^c$ is not $X$, or $\Omega \setminus X$). Prove your answer. \#4: Using the Method of Inclusion -Exclusion, count how many hands of 5 cards have at least one ace. You need to determine what the events $A_i$ should be. Do not find the answer by using the Law of Total Probability and complements (though you should use this to check your answer). \#5: We are going to divide 15 identical cookies among four people. How many ways are there to divide the cookies if all that matters is how many cookies a person receives? Redo this problem but now only consider divisions of the cookies where person $i$ gets at least $i$ cookies (thus person 1 must get at least one cookie, and so on). \#6: Redo the previous problem (15 identical cookies and 4 people), but with the following constraints: each person gets at most 10 cookies (it's thus possible some people get no cookies). \#7: Find a discrete random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for some $x$ between 17 and 17.01. \#8: Find a continuous random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for all $x$ between 17 and 17.01. \#9: Let $X$ be a continuous random variable with pdf $f_X$ satisfying $f_X(x) = f_X(-x)$. What can you deduce about $F_X$, the cdf? \#10: Find if you can, or say why you cannot, the first five Taylor coefficients of (a) $\log(1-u)$ at $u=0$; (b) $\log(1-u^2)$ at $u=0$; (c) $x \sin(1/x)$ at $x=0$. \#11: Let $X$ be a continuous random variable. (a) Prove $F_X$ is a non-decreasing function; this means $F_X(x) \le F_X(y)$ if $x < y$. (b) Let $U$ be a random variable with cdf $F_U(x) = 0$ if $u<0$, $F_U(x) = x$ if $0 < x < 1$, and $F_U(x) = 1$ if $1 < x$. Let $F$ be any continuous function such that $F$ is strictly increasing and the limit as $x$ approaches negative infinity of $F(x)$ is 0 and the limit as $x$ approaches positive infinity is 1. Prove $Y = F^{-1}(U)$ is a random variable with cdf $F$.
Week 4: February 24 - 28
- Videos: Tuesday: https://youtu.be/xMAzWda4yZU Thursday: https://youtu.be/e5H3XxPxdJk
- Optional Video: 2019: Card Shuffling (Guest Lecturer Francis Su): https://youtu.be/L_VTwdyyzUU
- Read: Chapter 5 and 6 of my textbook.
- Homework due Thursday March 6: #1: Imagine we have a deck with s suits and N cards in each suit. We play the game Aces Up, except now we put down s cards on each turn. What is the probability that the final s cards are all in different suits? Write a computer program to simulate 1,000,000 deals and compare your observed probability with your theoretical prediction; it is fine to just do the program for s=4 and N=13 (a standard hand); you may earn 15 out of 10 points if you write general code for general s, N. #2: Consider all generalized games of Aces Up with C cards in s suits with N cards in a suit; thus C = sN. What values of s and N give us the greatest chance of all the cards being in different suits? Of being in the same suit? #3: The double factorial is defined as the product of every other integer down to 1 or 2; thus 6!! = 6 * 4 * 2 while 7!! = 7 * 5 * 3 * 1. One can write (2n-1)!! as a! / (b^c d!) where a, b, c and d depend on n; find this elegant formula! Hint: b turns out to be a constant, taking the same value for all n. #4: A regular straight is five cards (not necessarily in the same suit) of five consecutive numbers; aces may be high or low, but we are not allowed to wrap around. A kangaroo straight differs in that the cards now differ by 2 (for example, 4 6 8 10 Q). What is the probability someone is dealt a kangaroo straight in a hand of five cards? #5: A prisoner is given an interesting chance for parole. He's blindfolded and told to choose one of two bags; once he does, he is to reach in and pull out a marble. Each bag has 25 red and 25 black marbles, and the marbles all feel the same. If he pulls out a red marble he is set free; if it's a black, his parole is denied. What is his chance of winning parole? #6: The set-up is similar to the previous problem, except now the prisoner is free to distribute the marbles among the two bags however he wishes, so long as all the marbles are distributed. He's blindfolded again, chooses a bag at random again, and then a marble. What is the best probability he can do for being set free? While you can get some points for writing down the correct answer, to receive full credit you must prove your answer is optimal!
Week 3: February 17 - 21, 2025
- Read: Chapters 4 and Chapter 5 of my textbook
- HW: Due February 20: #1: Section 1.2: Modify the basketball game so that there are 2015 players, numbered 1, 2, ..., 2015. Player i always gets a basket with probability 1/2ⁱ. What is the probability the first player wins? #2: Section 1.2: Is the answer for Example 1.2.1 consistent with what you would expect in the limit as c tends to minus infinity? #3: Section 1.2: Compute the first 42 terms of 1/998999 and comment on what you find; you may use a computer. #4: Section 2.2.1: Find sets A and B such that |A| = |B|, A is a subset of the real line and B is a subset of the plane (i.e., R²) but is not a subset of any line. #5: Section 2.2.1: Write at most a paragraph on the continuum hypothesis (you may use Wikipedia or any source to look it up). #6: Section 2.2.2: Give an example of an open set, a closed set, and a set that is neither open nor closed (you may not use the examples in the book); say a few words justifying your answer. #7: Section 2.3: Give another proof that the probability of the empty set is zero. #8: Find the probability of rolling exactly k sixes when we roll five fair die for k = 0, 1, ..., 5. Compare the work needed here to the complement approach in the book. #9: If f and g are differentiable functions, prove the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). Emphasize where you add zero.
- Also use this time to make sure you can do simple, basic coding. I don't care what language you use (Mathematica, R, Python, Fortran, ...), but you should be comfortable doing simple assignments. I'll post a list of basic problems you should be able to do. If you want to learn Mathematica, I have a template online and a YouTube tutorial. Just go to http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm (note you'll also get links to using LaTeX). Here is a Mathematica file of code for problems in the book (with comments), as well as some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf). If you have trouble with any of these, please let me know -- you are responsible for making sure you can write code to solve at least half of these problems. There are also videos online from introducing coding in my problem solving class: 2017 here: https://youtu.be/e8I5alerkOw and 2018 here: https://youtu.be/e8I5alerkOw (code here: http://web.williams.edu/Mathematics/sjmiller/public_html/331Fa18/mathematicaprograms/math331IntroCoding2018.nb).
- Watch at least 2 of the following videos (note the first is included in the first mathematical modeling).
- Roulette: Double Plus Ungood: https://youtu.be/Esa2TYwDmwA
- German Tank Problem: pdf (video: https://youtu.be/1N2IhpifwAk).
- Probability and Mathematical Modeling: I: (slides online here): https://youtu.be/T-35mhZioeo)

Week 2: February 10 - 14, 2025

Read: Chapter 2 and Chapter 3 of my textbook
Videos: Tuesday: https://youtu.be/r2vIDHJOoSQ Thursday: https://youtu.be/aByMKJ8NPEE
HW: Due February 20: #1: Section 1.2: Modify the basketball game so that there are 2015 players, numbered 1, 2, ..., 2015. Player i always gets a basket with probability 1/2ⁱ. What is the probability the first player wins? #2: Section 1.2: Is the answer for Example 1.2.1 consistent with what you would expect in the limit as c tends to minus infinity? #3: Section 1.2: Compute the first 42 terms of 1/998999 and comment on what you find; you may use a computer. #4: Section 2.2.1: Find sets A and B such that |A| = |B|, A is a subset of the real line and B is a subset of the plane (i.e., R²) but is not a subset of any line. #5: Section 2.2.1: Write at most a paragraph on the continuum hypothesis (you may use Wikipedia or any source to look it up). #6: Section 2.2.2: Give an example of an open set, a closed set, and a set that is neither open nor closed (you may not use the examples in the book); say a few words justifying your answer. #7: Section 2.3: Give another proof that the probability of the empty set is zero. #8: Find the probability of rolling exactly k sixes when we roll five fair die for k = 0, 1, ..., 5. Compare the work needed here to the complement approach in the book. #9: If f and g are differentiable functions, prove the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). Emphasize where you add zero.

Week 1: February 5, 2025
- Read: Chapter 1 of the textbook and the graduation speech here: https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/
- Homework: To be emailed to me by 10pm on Sunday, February 9: Email me a short note on what you want to get out of this course, and what lesson you learned from the graduation speech: https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/ . Full credit, 20/20, if you answer both questions on time. If you've already said what you want from a previous course with me, please briefly repeat.
- click here for first day slides click here for first day's handout first day's video: https://youtu.be/bSdLy3-BVvQ

FALL 2021

Week 14: December 6-10

Videos: Mon: https://youtu.be/qQjVN2C0gTg

Week 13: November 29 - Dec 3

Videos: Mon: https://youtu.be/0t_I5j598vQ (slides) Wed: Fri:
HW: Make sure you understand all problems on the midterm. Make sure you are able to do simple coding.
HW: Due Monday December 4: From the handout https://web.williams.edu/Mathematics/sjmiller/public_html/105Sp10/handouts/MethodLeastSquares.pdf do Exercise 3.9.

Week 12: November 22

Videos: Mon:
Remember to do midterm.... Due at the start of class on Monday November 29th (but of course feel free to send earlier).
HW: Due Monday December 4: From the handout https://web.williams.edu/Mathematics/sjmiller/public_html/105Sp10/handouts/MethodLeastSquares.pdf do Exercise 3.9.

Week 11: November 15 - 19

Videos:
Homework: Make sure you are caught up on all the material, prepare for the midterm

Week 10: Nov 8 to 12:

Read: Review Chapters 19 and 20, read Chapter 21
Videos: Mon: https://youtu.be/eBcKGUSB_vI (slides) Wed: https://youtu.be/fY7teGfgxsY (slides) Fri: https://youtu.be/i0NX8vb9rWU (slides)
Homework: Due Friday November 12: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.

Week 9: Nov 1 - 5 (no class on Fri Nov 5)

Read: Read up to and including Chapter 20
Video: Mon: https://youtu.be/qW-3bHAwdPU (slides) Wed: Fri:
Homework: Due Friday November 12: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.

Week 8: Oct 21-25 (Possible Asynchronous Class Friday)
- Read: Chapters 15 to 18
- Video: Mon: https://youtu.be/iltn3Oks-9k (slides) Wed:
- Homework: Monday, November 1: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 7: Oct 18 - 22
- Read: Chapters 15 to 18
- Video: Wed: Sick, lecture from 2019: Simpson's paradox, Ace of Hearts method: https://youtu.be/-on_cFHW66g (slides here) Fri: https://youtu.be/iJyoyUV2JWQ (slides here)
- Homework: Due MONDAY November 1: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 6: October 11 to 15 (no class on Monday b/c of Reading Period, no class on Friday b/c of take-home exam)
- Videos: Mon: Wed: Fri:
- Read: Make sure you have read through Chapter 11 (though I doubt we'll get through all of Chapter 11, have a strategic reserve of reading!)
- HW: Due Friday, October 29: #1: We toss $n$ fair coins. Every coin that lands on heads is tossed again. What is the probability density function for the number of heads after the second set of tosses (i.e., after we have retossed all the coins that landed on heads)? If you want, imagine you have left the room and return after all the tossing is done; what is the pdf for the number of heads you see? #2: Is there a $C$ such that $f(x) = C Exp(-x - Exp(-x))$ is a probability density function? Here $-\infty < x < \infty$. #3: Let $X$ be a discrete random variable. Prove or disprove: $E[1/X] = 1/E[X]$. #4: Let $X_1, \dots, X_n$ be independent, identically distributed random variables that have zero probability of taking on a non-positive value. Prove $E[(X_1 + \cdots + X_m) / (X_1 + \cdots + X_n)] = m/n$ for $1 \le m \le n$. Does this result seem surprising? Write a computer program to investigate when the random variables are drawn from a uniform distribution on [0,1]. #5: Let $X$ and $Y$ be two continuous random variables with densities $f_X$ and $f_Y$. (a) For what $c$ is $c f_X(x) + (1-c) f_Y(x)$ a density? (b) Can there be a continuous random variable with pdf equal to $f_X(x) f_Y(x)$? #6: The standard normal has density $\frac1{\sqrt{2\pi}} Exp(-x^2/2)$ (this means this integrates to 1). Find the first four moments. #7: Find the errorS in the following code:
- hoops[p_, q_, need_, num_] := Module[{},
  birdwin = 0;
  For[n = 1, n <= num,
  {
      If[Mod[n, num/10] == 0, Print["We have done ", 100. n/num, "%."]];
      birdbasket = 0;
      magicbasket = 0;
      While[birdbasket < need || magicbasket < need,
          {
              If[Random[] <= p, birdbasket = birdbasket + 1];
              If[Random[] <= q, magicbasket = magicbasket + 1];
          }]; (* end of while loop *)
      If[birdbasket == need, birdwin == birdwin + 1];
  }]; (* end of for loop *)
  Print["Bird wins ", 100. birdwin/num, "%."];
  Print["Magic wins ", 100. - 100. birdwin/num, "%."];
  ];
  
  hoops[.32, .33, 5, 100]

Week 5: October 4 - 8
- Read: Chapters 7 to 10.
- Video: Mon: Wed: Fri:
- HW \#4: Due Friday, Oct 8: Note Mathematical Induction might be useful for some of these problems. \#1: Let $\{A_n\}_{n=1}^\infty$ be a countable sequence of events such that for each $n$, ${\rm Prob}(A_n) = 1$. Prove the probability of the intersection of all the $A_n$'s is 1. \#2: Prove the number of ways to match $2n$ people into $n$ pairs of 2 is $(2n-1)!!$ (recall the double factorial is the product of every other integer, continuing down to 2 or 1). \#3: Assume $0 < {\rm Prob}(X), {\rm Prob}(Y) < 1$ and $X$ and $Y$ are independent. Are $X^c$ and $Y^c$ independent? (Note $X^c$ is not $X$, or $\Omega \setminus X$). Prove your answer. \#4: Using the Method of Inclusion -Exclusion, count how many hands of 5 cards have at least one ace. You need to determine what the events $A_i$ should be. Do not find the answer by using the Law of Total Probability and complements (though you should use this to check your answer). \#5: We are going to divide 15 identical cookies among four people. How many ways are there to divide the cookies if all that matters is how many cookies a person receives? Redo this problem but now only consider divisions of the cookies where person $i$ gets at least $i$ cookies (thus person 1 must get at least one cookie, and so on). \#6: Redo the previous problem (15 identical cookies and 4 people), but with the following constraints: each person gets at most 10 cookies (it's thus possible some people get no cookies). \#7: Find a discrete random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for some $x$ between 17 and 17.01. \#8: Find a continuous random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for all $x$ between 17 and 17.01. \#9: Let $X$ be a continuous random variable with pdf $f_X$ satisfying $f_X(x) = f_X(-x)$. What can you deduce about $F_X$, the cdf? \#10: Find if you can, or say why you cannot, the first five Taylor coefficients of (a) $\log(1-u)$ at $u=0$; (b) $\log(1-u^2)$ at $u=0$; (c) $x \sin(1/x)$ at $x=0$. \#11: Let $X$ be a continuous random variable. (a) Prove $F_X$ is a non-decreasing function; this means $F_X(x) \le F_X(y)$ if $x < y$. (b) Let $U$ be a random variable with cdf $F_U(x) = 0$ if $u<0$, $F_U(x) = x$ if $0 < x < 1$, and $F_U(x) = 1$ if $1 < x$. Let $F$ be any continuous function such that $F$ is strictly increasing and the limit as $x$ approaches negative infinity of $F(x)$ is 0 and the limit as $x$ approaches positive infinity is 1. Prove $Y = F^{-1}(U)$ is a random variable with cdf $F$.
Week 4: September 27 - October 1: no in-person class on Friday
- Videos 2019: Mon: Card Shuffling (Guest Lecturer Francis Su): https://youtu.be/L_VTwdyyzUU Wed: https://youtu.be/Id0h1Nawh9Q Fri: https://youtu.be/lJKSKEUZ69k
- Read: Chapter 5 and 6 of my textbook.
- Homework due Friday October 1: #1: Imagine we have a deck with s suits and N cards in each suit. We play the game Aces Up, except now we put down s cards on each turn. What is the probability that the final s cards are all in different suits? Write a computer program to simulate 1,000,000 deals and compare your observed probability with your theoretical prediction; it is fine to just do the program for s=4 and N=13 (a standard hand); you may earn 15 out of 10 points if you write general code for general s, N. #2: Consider all generalized games of Aces Up with C cards in s suits with N cards in a suit; thus C = sN. What values of s and N give us the greatest chance of all the cards being in different suits? Of being in the same suit? #3: The double factorial is defined as the product of every other integer down to 1 or 2; thus 6!! = 6 * 4 * 2 while 7!! = 7 * 5 * 3 * 1. One can write (2n-1)!! as a! / (b^c d!) where a, b, c and d depend on n; find this elegant formula! Hint: b turns out to be a constant, taking the same value for all n. #4: A regular straight is five cards (not necessarily in the same suit) of five consecutive numbers; aces may be high or low, but we are not allowed to wrap around. A kangaroo straight differs in that the cards now differ by 2 (for example, 4 6 8 10 Q). What is the probability someone is dealt a kangaroo straight in a hand of five cards? #5: A prisoner is given an interesting chance for parole. He's blindfolded and told to choose one of two bags; once he does, he is to reach in and pull out a marble. Each bag has 25 red and 25 black marbles, and the marbles all feel the same. If he pulls out a red marble he is set free; if it's a black, his parole is denied. What is his chance of winning parole? #6: The set-up is similar to the previous problem, except now the prisoner is free to distribute the marbles among the two bags however he wishes, so long as all the marbles are distributed. He's blindfolded again, chooses a bag at random again, and then a marble. What is the best probability he can do for being set free? While you can get some points for writing down the correct answer, to receive full credit you must prove your answer is optimal!

Week 3: September 20-24
- No written homework due this Friday! Instead use the time to build up your strategic reserve in the book. We will not cover most of Chapter 3 in class -- read the material and if there are calculations you are having trouble with or want to see in class, let me know and I'll do. Start reading Chapter Four. Monday's class will be a quick run of the Chapter Three material. Later we'll talk about some applications of probability to mathematical modeling. There will not be reading assigned for this; the purpose of this is to (1) quickly show you how useful probability can be, and (2) give you a sense of the tools and techniques we'll see later in the semester, and (3) give you plenty of time to read ahead and build up your strategic reserve (if you don't take advantage of this you'll have some painful weeks down the road!).
- Also use this time to make sure you can do simple, basic coding. I don't care what language you use (Mathematica, R, Python, Fortran, ...), but you should be comfortable doing simple assignments. I'll post a list of basic problems you should be able to do. If you want to learn Mathematica, I have a template online and a YouTube tutorial. Just go to http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm (note you'll also get links to using LaTeX). Here is a Mathematica file of code for problems in the book (with comments), as well as some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf). If you have trouble with any of these, please let me know -- you are responsible for making sure you can write code to solve at least half of these problems. There are also videos online from introducing coding in my problem solving class: 2017 here: https://youtu.be/e8I5alerkOw and 2018 here: https://youtu.be/e8I5alerkOw (code here: http://web.williams.edu/Mathematics/sjmiller/public_html/331Fa18/mathematicaprograms/math331IntroCoding2018.nb).

Week 2: September 13-17
- Read: Chapter 2 and Chapter 3 of my textbook
- Videos: Monday: http://youtu.be/dmI9d-w-bM4 Wed: http://youtu.be/Mg7xZqolBKE Fri: http://youtu.be/0Rp8KgWmLi0 (from last iteration of course)
- HW: Due September 17: #1: Section 1.2: Modify the basketball game so that there are 2015 players, numbered 1, 2, ..., 2015. Player i always gets a basket with probability 1/2ⁱ. What is the probability the first player wins? #2: Section 1.2: Is the answer for Example 1.2.1 consistent with what you would expect in the limit as c tends to minus infinity? #3: Section 1.2: Compute the first 42 terms of 1/998999 and comment on what you find; you may use a computer. #4: Section 2.2.1: Find sets A and B such that |A| = |B|, A is a subset of the real line and B is a subset of the plane (i.e., R²) but is not a subset of any line. #5: Section 2.2.1: Write at most a paragraph on the continuum hypothesis (you may use Wikipedia or any source to look it up). #6: Section 2.2.2: Give an example of an open set, a closed set, and a set that is neither open nor closed (you may not use the examples in the book); say a few words justifying your answer. #7: Section 2.3: Give another proof that the probability of the empty set is zero. #8: Find the probability of rolling exactly k sixes when we roll five fair die for k = 0, 1, ..., 5. Compare the work needed here to the complement approach in the book. #9: If f and g are differentiable functions, prove the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). Emphasize where you add zero.

Week 1: September 10, 2021
- Read: Chapter 1 of the textbook and the graduation speech here: https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/
- Homework: To be emailed to me by 10pm on Sunday, September 12: Email me a short note on what you want to get out of this course, and what lesson you learned from the graduation speech: https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/ . Full credit, 20/20, if you answer both questions on time. If you've already said what you want from a previous course with me, please briefly repeat. Thanks!
- click here for first day slides click here for first day's handout first day's video: https://youtu.be/aD7tIO0CGyg (these are from 2021)

THE PROBLEMS BELOW ARE FROM THE 2019 ITERATION

Week 14: December 2 - 6

Videos: Mon: https://youtu.be/WJdmu2vleTY

Week 13: November 25
- Videos: Mon: https://youtu.be/WX8PScrb8oM
Week 12: November 18-22
- Videos: Mon: http://youtu.be/_L72XJXxGCc Wed: https://youtu.be/4m77G15eINk Fri: pdf (video: https://youtu.be/nZJUbwC_bTw).
- HW: review material from Midterm II
- HW: Due Wednesday December 1: From the handout https://web.williams.edu/Mathematics/sjmiller/public_html/105Sp10/handouts/MethodLeastSquares.pdf do Exercise 3.9.

Week 11: November 11-15: No Class (Takehome exam)
- Videos; Mon: https://youtu.be/vmBUx0VhoEQ Wed: http://youtu.be/Y7NppJoRoxQ Fri: No Class
- Homework: Due Friday November 15: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.

Week 10: Nov 4 to 8: Midterm 2 due by start of class Mon Nov 18th
- Read: Review Chapters 19 and 20, read Chapter 21
- Videos: Mon: https://youtu.be/_-aGxDkNLHY Wed: https://youtu.be/kVBIVl9uDTU Fri: https://youtu.be/GxohirsuMfM
- Homework: Due Friday November 15: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.
Week 9: Oct 28 - Nov 1: Second midterm will be given in the near future....
- Read: Read up to and including Chapter 20
- Video: Mon: https://youtu.be/N0ye_W-2MD4 Wed: https://youtu.be/X9Ujt8oicAI Fri: https://youtu.be/vuKCrS2on9Q
- Homework: Monday, November 4: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 8: Oct 21-25 (No Class Friday)
- Read: Chapters 15 to 18
- Video: Mon: https://youtu.be/-36UUC_oPUs Wed:
- Homework: Monday, November 4: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 7: Oct 14-18 (No Class Monday - Reading Period)
- Read: Chapters 15 to 18
- Video: Wed: https://youtu.be/-on_cFHW66g (slides here) Fri: https://youtu.be/WdITkk5zac0
- Homework: Due MONDAY November 1: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 6: October 7 to 11, 2019
- Videos: Mon: https://youtu.be/6Yr5rTSoQrQ Wed: https://youtu.be/9iqLKlL-KHY (slides here) Fri:
- Read: Make sure you have read through Chapter 11 (though I doubt we'll get through all of Chapter 11, have a strategic reserve of reading!)
- HW: Due Friday, October 18: #1: We toss $n$ fair coins. Every coin that lands on heads is tossed again. What is the probability density function for the number of heads after the second set of tosses (i.e., after we have retossed all the coins that landed on heads)? If you want, imagine you have left the room and return after all the tossing is done; what is the pdf for the number of heads you see? #2: Is there a $C$ such that $f(x) = C Exp(-x - Exp(-x))$ is a probability density function? Here $-\infty < x < \infty$. #3: Let $X$ be a discrete random variable. Prove or disprove: $E[1/X] = 1/E[X]$. #4: Let $X_1, \dots, X_n$ be independent, identically distributed random variables that have zero probability of taking on a non-positive value. Prove $E[(X_1 + \cdots + X_m) / (X_1 + \cdots + X_n)] = m/n$ for $1 \le m \le n$. Does this result seem surprising? Write a computer program to investigate when the random variables are drawn from a uniform distribution on [0,1]. #5: Let $X$ and $Y$ be two continuous random variables with densities $f_X$ and $f_Y$. (a) For what $c$ is $c f_X(x) + (1-c) f_Y(x)$ a density? (b) Can there be a continuous random variable with pdf equal to $f_X(x) f_Y(x)$? #6: The standard normal has density $\frac1{\sqrt{2\pi}} Exp(-x^2/2)$ (this means this integrates to 1). Find the first four moments. #7: Find the errorS in the following code:
- hoops[p_, q_, need_, num_] := Module[{},
  birdwin = 0;
  For[n = 1, n <= num,
  {
      If[Mod[n, num/10] == 0, Print["We have done ", 100. n/num, "%."]];
      birdbasket = 0;
      magicbasket = 0;
      While[birdbasket < need || magicbasket < need,
          {
              If[Random[] <= p, birdbasket = birdbasket + 1];
              If[Random[] <= q, magicbasket = magicbasket + 1];
          }]; (* end of while loop *)
      If[birdbasket == need, birdwin == birdwin + 1];
  }]; (* end of for loop *)
  Print["Bird wins ", 100. birdwin/num, "%."];
  Print["Magic wins ", 100. - 100. birdwin/num, "%."];
  ];
  
  hoops[.32, .33, 5, 100]

Week 5: Sept 30 - Oct 4, 2019: No physical class on Wed (will post a video)
- Read: Chapters 7 to 10.
- Video: Mon: https://youtu.be/sn-hypa9tRY (coding: https://youtu.be/sSgjBysixdQ; code file here, pdf here) Wed: https://youtu.be/fCQlNOiHolU Fri:
- HW \#4: Due Friday, Oct 4: Note Mathematical Induction might be useful for some of these problems. \#1: Let $\{A_n\}_{n=1}^\infty$ be a countable sequence of events such that for each $n$, ${\rm Prob}(A_n) = 1$. Prove the probability of the intersection of all the $A_n$'s is 1. \#2: Prove the number of ways to match $2n$ people into $n$ pairs of 2 is $(2n-1)!!$ (recall the double factorial is the product of every other integer, continuing down to 2 or 1). \#3: Assume $0 < {\rm Prob}(X), {\rm Prob}(Y) < 1$ and $X$ and $Y$ are independent. Are $X^c$ and $Y^c$ independent? (Note $X^c$ is not $X$, or $\Omega \setminus X$). Prove your answer. \#4: Using the Method of Inclusion -Exclusion, count how many hands of 5 cards have at least one ace. You need to determine what the events $A_i$ should be. Do not find the answer by using the Law of Total Probability and complements (though you should use this to check your answer). \#5: We are going to divide 15 identical cookies among four people. How many ways are there to divide the cookies if all that matters is how many cookies a person receives? Redo this problem but now only consider divisions of the cookies where person $i$ gets at least $i$ cookies (thus person 1 must get at least one cookie, and so on). \#6: Redo the previous problem (15 identical cookies and 4 people), but with the following constraints: each person gets at most 10 cookies (it's thus possible some people get no cookies). \#7: Find a discrete random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for some $x$ between 17 and 17.01. \#8: Find a continuous random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for all $x$ between 17 and 17.01. \#9: Let $X$ be a continuous random variable with pdf $f_X$ satisfying $f_X(x) = f_X(-x)$. What can you deduce about $F_X$, the cdf? \#10: Find if you can, or say why you cannot, the first five Taylor coefficients of (a) $\log(1-u)$ at $u=0$; (b) $\log(1-u^2)$ at $u=0$; (c) $x \sin(1/x)$ at $x=0$. \#11: Let $X$ be a continuous random variable. (a) Prove $F_X$ is a non-decreasing function; this means $F_X(x) \le F_X(y)$ if $x < y$. (b) Let $U$ be a random variable with cdf $F_U(x) = 0$ if $u<0$, $F_U(x) = x$ if $0 < x < 1$, and $F_U(x) = 1$ if $1 < x$. Let $F$ be any continuous function such that $F$ is strictly increasing and the limit as $x$ approaches negative infinity of $F(x)$ is 0 and the limit as $x$ approaches positive infinity is 1. Prove $Y = F^{-1}(U)$ is a random variable with cdf $F$.
Week 4: September 23-27: Guest Lecturer Mon Sept 23
- Videos: Mon: Card Shuffling (Guest Lecturer Francis Su): https://youtu.be/L_VTwdyyzUU Wed: https://youtu.be/Id0h1Nawh9Q Fri: https://youtu.be/lJKSKEUZ69k
- Read: Chapter 5 and 6 of my textbook.
- Homework due Friday September 27: #1: Imagine we have a deck with s suits and N cards in each suit. We play the game Aces Up, except now we put down s cards on each turn. What is the probability that the final s cards are all in different suits? Write a computer program to simulate 1,000,000 deals and compare your observed probability with your theoretical prediction; it is fine to just do the program for s=4 and N=13 (a standard hand); you may earn 15 out of 10 points if you write general code for general s, N. #2: Consider all generalized games of Aces Up with C cards in s suits with N cards in a suit; thus C = sN. What values of s and N give us the greatest chance of all the cards being in different suits? Of being in the same suit? #3: The double factorial is defined as the product of every other integer down to 1 or 2; thus 6!! = 6 * 4 * 2 while 7!! = 7 * 5 * 3 * 1. One can write (2n-1)!! as a! / (b^c d!) where a, b, c and d depend on n; find this elegant formula! Hint: b turns out to be a constant, taking the same value for all n. #4: A regular straight is five cards (not necessarily in the same suit) of five consecutive numbers; aces may be high or low, but we are not allowed to wrap around. A kangaroo straight differs in that the cards now differ by 2 (for example, 4 6 8 10 Q). What is the probability someone is dealt a kangaroo straight in a hand of five cards? #5: A prisoner is given an interesting chance for parole. He's blindfolded and told to choose one of two bags; once he does, he is to reach in and pull out a marble. Each bag has 25 red and 25 black marbles, and the marbles all feel the same. If he pulls out a red marble he is set free; if it's a black, his parole is denied. What is his chance of winning parole? #6: The set-up is similar to the previous problem, except now the prisoner is free to distribute the marbles among the two bags however he wishes, so long as all the marbles are distributed. He's blindfolded again, chooses a bag at random again, and then a marble. What is the best probability he can do for being set free? While you can get some points for writing down the correct answer, to receive full credit you must prove your answer is optimal!

Week 3: September 16-20 (remember no physical class on Monday)
- No written homework! Instead use the time to build up your strategic reserve in the book. We will not cover most of Chapter 3 in class -- read the material and if there are calculations you are having trouble with or want to see in class, let me know and I'll do. Start reading Chapter Four. Monday's class will be a quick run of the Chapter Three material. Later we'll talk about some applications of probability to mathematical modeling. There will not be reading assigned for this; the purpose of this is to (1) quickly show you how useful probability can be, and (2) give you a sense of the tools and techniques we'll see later in the semester, and (3) give you plenty of time to read ahead and build up your strategic reserve (if you don't take advantage of this you'll have some painful weeks down the road!).
- Also use this time to make sure you can do simple, basic coding. I don't care what language you use (Mathematica, R, Python, Fortran, ...), but you should be comfortable doing simple assignments. I'll post a list of basic problems you should be able to do. If you want to learn Mathematica, I have a template online and a YouTube tutorial. Just go to http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm (note you'll also get links to using LaTeX). Here is a Mathematica file of code for problems in the book (with comments), as well as some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf). If you have trouble with any of these, please let me know -- you are responsible for making sure you can write code to solve at least half of these problems. There are also videos online from introducing coding in my problem solving class: 2017 here: https://youtu.be/e8I5alerkOw and 2018 here: https://youtu.be/e8I5alerkOw (code here: http://web.williams.edu/Mathematics/sjmiller/public_html/331Fa18/mathematicaprograms/math331IntroCoding2018.nb).

Week 2: September 9-13, 2019
- Read: Chapter 2 and Chapter 3 of my textbook
- Videos: Monday: https://youtu.be/4fzd-IRV2cE Wed: https://youtu.be/2agUkFQJtnU Fri: https://youtu.be/qiek1ZM_KLE (2018 version) and supplemental on infinities, generating functions and differentiating identities: https://youtu.be/TQ20q4yZsho
- HW: Due Friday September 13, 2019: #1: Section 1.2: Modify the basketball game so that there are 2015 players, numbered 1, 2, ..., 2015. Player i always gets a basket with probability 1/2ⁱ. What is the probability the first player wins? #2: Section 1.2: Is the answer for Example 1.2.1 consistent with what you would expect in the limit as c tends to minus infinity? #3: Section 1.2: Compute the first 42 terms of 1/998999 and comment on what you find; you may use a computer. #4: Section 2.2.1: Find sets A and B such that |A| = |B|, A is a subset of the real line and B is a subset of the plane (i.e., R²) but is not a subset of any line. #5: Section 2.2.1: Write at most a paragraph on the continuum hypothesis (you may use Wikipedia or any source to look it up). #6: Section 2.2.2: Give an example of an open set, a closed set, and a set that is neither open nor closed (you may not use the examples in the book); say a few words justifying your answer. #7: Section 2.3: Give another proof that the probability of the empty set is zero. #8: Find the probability of rolling exactly k sixes when we roll five fair die for k = 0, 1, ..., 5. Compare the work needed here to the complement approach in the book. #9: If f and g are differentiable functions, prove the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). Emphasize where you add zero.

Week 1: September 6, 2019
- Read: Chapter 1 of the textbook and the graduation speech here: http://www.graduationwisdom.com/speeches/0018-uslan.htm
- Homework: To be emailed to me by 10pm on Sunday, September 8: Email me a short note on what you want to get out of this course, and what lesson you learned from the graduation speech. Full credit, 20/20, if you answer both questions on time. If you've already said what you want from a previous course with me, please briefly repeat. Thanks!
- click here for first day slides click here for first day's handout first day's video: https://youtu.be/HIv-HCVUfHM

THE PROBLEMS BELOW ARE FROM THE 2018 ITERATION

Week 12: Nov 26 to 30:
- Homework: Review coding, prepare for exams, do HW exam ....

Week 12: Nov 19: Thanksgiving:

Week 11: November 12-16:
- Homework:
  - Due Friday November 16 (email to me, do on a separate sheet than problem below): Find any math research paper or expository paper which uses probability and write an at most one page summary (preferably in TeX). As you continue in your careers, you are going to need to read technical papers and summarize them to your superiors / colleagues / clients; this is thus potentially a very useful exercise. Make sure you describe clearly what the point of the paper is, what techniques are used to study the problem, what applications there are (if any). Below is a sample review from MathSciNet; if you would like to see more, you can go to their homepage or ask me and I'll pass along many of the ones I've written. I've chosen this one as it's related to a paper on randomly shuffling cards: Bayer, Dave and Diaconis, Persi, Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2 (1992), no. 2, 294--313.
    - Rarely does a new mathematical result make both the New York Times and the front page of my local paper, and even more rarely is your reviewer asked to speak on commercial radio about a result, but such activity was caused by the preprint of this paper. In layman's terms, it says you should shuffle a deck of cards seven times before playing. More technically, the usual way people shuffle is called a riffle shuffle, and a natural mathematical model of a random shuffle is to assume all possible riffle shuffles are equally likely. With this model one can ask how close is $k$ shuffles of an n-card deck to the uniform distribution on all $n!$ permutations, where `close' is measured by variation distance. It was previously known that, as $n \to \infty$, one needs $k(n) \sim 3 2 \log_2 n$ shuffles to get close to uniform. This paper gives an elegant and careful treatment based on an explicit formula for the exact distance $d(k, n)$to uniformity. To quote the abstract: `Key ingredients are the analysis of a card trick and the determination of the idempotents of a natural commutative subalgebra in the symmetric group algebra.' -- Reviewed by David J. Aldous
  - Due Friday November 16: Assume two people randomly move in tic - tac - toe; what is the probability each wins (or that there is a draw)? Solve this both through simulation (i.e., writing a program) and theoretically. The number of possible games is small enough (9! is about 362, 880 ) that we can just do the theory by brute force computation. If we had a bigger board it would be very different and then simulations would be the way to go. THIS IS WORTH 40 POINTS, OR FOUR PROBLEMS; SOLVING THEORETICALLY IS WORTH TWO, WRITING THE CODE IS WORTH TWO. Thus if you are using a homework exemption, each part requires two problems.

Week 10: Nov 5 to 9: Midterm 2 due by start of class Mon Nov 5th, guest speaker Fri Nov 9 (Class in Griffen 6)
- Read: Review Chapters 19 and 20, read Chapter 21
- Homework: Due Friday November 9: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.
Week 9: Oct 29 - Nov 2: Second take-home midterm due start of class on Mon Nov 5th
- Read: Have already read 17 and 18; continue reading and do 19 and 20
- Homework: Due Friday November 9: #1: 10% of the numbers on a list are 15, 20% are 25, and the rest are 50. What is the average? #2: All 100 numbers in a list are non-negative and the average is 2. Prove that at most 25 exceed 8. #3: $A$ and $B$ independent events with indicator random variables $I_A$ and $I_B$; thus $I_A(x) = 1$ with probability ${\rm Pr}(A)$ and is 0 with probability $1-{\rm Pr}(A)$. (a) What is the distribution of $(I_A+I_B)^2$? (b) What is $E[(I_A+I_B)^2]$? #4: Consider a random variable $X$ with expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for ${\rm Pr}(X \ge 20)$. (b) Could $X$ be a binomial random variable? #5: Suppose average family income is $10,000. (a) Find an upper bound for the percentage of families with income over $50,000. (b) Redo (a) but with the added knowledge that the standard deviation is $8,000. #6: (a) Let $X$ be a random variable with $0 \le X \le 1$ and $E[X]=\mu$. Show that $0\le \mu \le 1$ and $0 \le {\rm Var}(X) \le \mu(1-\mu) \le 1/4$. (b) Generalize and consider the case $a \le X \le b$. (c) Assume $0 \le X \le 9$. Find a random variable where the variance is as large as possible.
Week 8: Oct 22-26 (No Class Friday)
- Read: Chapters 15 to 18
- Homework: Due Friday October 26: #1: Calculate the second and third moments of X when X ~ Bin(n,p) (a binomial random variable with parameters n and p). #2: We toss N coins (each of which is heads with probability p), where the number N is drawn from a Poisson random variable with parameter lambda. Let X denote the number of heads. What is the probability density function of X? Justify your answer. #3: Find the probability density function of Y when Y = exp(X) for X ~ N(0,1). #4: Each box of cereal is equally likely to have exactly one of a set of c prizes. Thus, every time you open a box you have a 1/c chance of getting prize 1, a 1/c chance of getting prize 2, .... How many boxes to you expect to have to open before you have at least one of each of the c prizes? #5: Let $X_1, ..., X_n$ be independent Bernoulli random variables where X_k ~ Bern(p_k) (you can think of this as n independent coin tosses, where coin k is heads with probability p). If $Y = X_1 + ... + X_n$, what is the mean and what is the variance of $Y$? Assume $p_1 + ... + p_n = \mu$; what choice or choices of the $p_k$'s lead to the variance of $Y$ being the largest possible? #6: State anything you learned or enjoyed in Arms' talk. One or two sentences suffice. #7: The kurtosis of a random variable X is defined by ${\rm kur}(X) := E[(X-\mu)^4] / \sigma^4$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The kurtosis measures how much probability we have in the tails. If $X \sim {\rm Poiss}(\lambda)$, find the kurtosis of $X$. #8: Consider a coin with probability $p$ of heads. Find the probability density function for $X_1$, where $X_1$ is how long we must wait before we get our first head. #9: Consider a coin with probability $p$ of heads. Find the probability density function for $X_2$, where $X_2$ is how long we must weight before we get our second head. #10: Alice, Bob and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6. What is the probability each of them wins? #11: Alice, Bob and Charlie are rolling a fair die in that order. What is the probability Alice is the first person to roll a 6, Bob is the second and Charlie is the third? #12: Alice, Bob and Charlie are still rolling the fair die. What is the probability that the first 6 is rolled by Alice, the second 6 by Bob and the third 6 by Charlie? #13: What are the mean and variance of a chi-square distribution with 2 degrees of freedom? If X ~ chi^2(2), what is the probability that X takes on a value at least twice its mean? What is the probability X takes on a value at most half of its mean?
Week 7: October 15 to 19, 2018
- Read: Read: Chapters 11 through 14
- HW: Due Friday, October 19: #1: We toss $n$ fair coins. Every coin that lands on heads is tossed again. What is the probability density function for the number of heads after the second set of tosses (i.e., after we have retossed all the coins that landed on heads)? If you want, imagine you have left the room and return after all the tossing is done; what is the pdf for the number of heads you see? #2: Is there a $C$ such that $f(x) = C Exp(-x - Exp(-x))$ is a probability density function? Here $-\infty < x < \infty$. #3: Let $X$ be a discrete random variable. Prove or disprove: $E[1/X] = 1/E[X]$. #4: Let $X_1, \dots, X_n$ be independent, identically distributed random variables that have zero probability of taking on a non-positive value. Prove $E[(X_1 + \cdots + X_m) / (X_1 + \cdots + X_n)] = m/n$ for $1 \le m \le n$. Does this result seem surprising? Write a computer program to investigate when the random variables are drawn from a uniform distribution on [0,1]. #5: Let $X$ and $Y$ be two continuous random variables with densities $f_X$ and $f_Y$. (a) For what $c$ is $c f_X(x) + (1-c) f_Y(x)$ a density? (b) Can there be a continuous random variable with pdf equal to $f_X(x) f_Y(x)$? #6: The standard normal has density $\frac1{\sqrt{2\pi}} Exp(-x^2/2)$ (this means this integrates to 1). Find the first four moments. #7: Find the errorS in the following code:
- hoops[p_, q_, need_, num_] := Module[{},
  birdwin = 0;
  For[n = 1, n <= num,
  {
      If[Mod[n, num/10] == 0, Print["We have done ", 100. n/num, "%."]];
      birdbasket = 0;
      magicbasket = 0;
      While[birdbasket < need || magicbasket < need,
          {
              If[Random[] <= p, birdbasket = birdbasket + 1];
              If[Random[] <= q, magicbasket = magicbasket + 1];
          }]; (* end of while loop *)
      If[birdbasket == need, birdwin == birdwin + 1];
  }]; (* end of for loop *)
  Print["Bird wins ", 100. birdwin/num, "%."];
  Print["Magic wins ", 100. - 100. birdwin/num, "%."];
  ];
  
  hoops[.32, .33, 5, 100]

Week 6: October 8 to 12, 2018
- Videos: Mon: Wed: Fri:
- Read: Make sure you have read through Chapter 11 (though I doubt we'll get through all of Chapter 11, have a strategic reserve of reading!)
- HW: Due Friday, October 19: #1: We toss $n$ fair coins. Every coin that lands on heads is tossed again. What is the probability density function for the number of heads after the second set of tosses (i.e., after we have retossed all the coins that landed on heads)? If you want, imagine you have left the room and return after all the tossing is done; what is the pdf for the number of heads you see? #2: Is there a $C$ such that $f(x) = C Exp(-x - Exp(-x))$ is a probability density function? Here $-\infty < x < \infty$. #3: Let $X$ be a discrete random variable. Prove or disprove: $E[1/X] = 1/E[X]$. #4: Let $X_1, \dots, X_n$ be independent, identically distributed random variables that have zero probability of taking on a non-positive value. Prove $E[(X_1 + \cdots + X_m) / (X_1 + \cdots + X_n)] = m/n$ for $1 \le m \le n$. Does this result seem surprising? Write a computer program to investigate when the random variables are drawn from a uniform distribution on [0,1]. #5: Let $X$ and $Y$ be two continuous random variables with densities $f_X$ and $f_Y$. (a) For what $c$ is $c f_X(x) + (1-c) f_Y(x)$ a density? (b) Can there be a continuous random variable with pdf equal to $f_X(x) f_Y(x)$? #6: The standard normal has density $\frac1{\sqrt{2\pi}} Exp(-x^2/2)$ (this means this integrates to 1). Find the first four moments. #7: Find the errorS in the following code:
- hoops[p_, q_, need_, num_] := Module[{},
  birdwin = 0;
  For[n = 1, n <= num,
  {
      If[Mod[n, num/10] == 0, Print["We have done ", 100. n/num, "%."]];
      birdbasket = 0;
      magicbasket = 0;
      While[birdbasket < need || magicbasket < need,
          {
              If[Random[] <= p, birdbasket = birdbasket + 1];
              If[Random[] <= q, magicbasket = magicbasket + 1];
          }]; (* end of while loop *)
      If[birdbasket == need, birdwin == birdwin + 1];
  }]; (* end of for loop *)
  Print["Bird wins ", 100. birdwin/num, "%."];
  Print["Magic wins ", 100. - 100. birdwin/num, "%."];
  ];
  
  hoops[.32, .33, 5, 100]

Week 5: Oct 1 - 5: (no class Friday, take home exam tentatively due Wednesday October 10th by the start of class)
- Read: Chapters 7 to 10.
- HW \#4: Due Friday, Oct 5: Note Mathematical Induction might be useful for some of these problems. \#1: Let $\{A_n\}_{n=1}^\infty$ be a countable sequence of events such that for each $n$, ${\rm Prob}(A_n) = 1$. Prove the probability of the intersection of all the $A_n$'s is 1. \#2: Prove the number of ways to match $2n$ people into $n$ pairs of 2 is $(2n-1)!!$ (recall the double factorial is the product of every other integer, continuing down to 2 or 1). \#3: Assume $0 < {\rm Prob}(X), {\rm Prob}(Y) < 1$ and $X$ and $Y$ are independent. Are $X^c$ and $Y^c$ independent? (Note $X^c$ is not $X$, or $\Omega \setminus X$). Prove your answer. \#4: Using the Method of Inclusion -Exclusion, count how many hands of 5 cards have at least one ace. You need to determine what the events $A_i$ should be. Do not find the answer by using the Law of Total Probability and complements (though you should use this to check your answer). \#5: We are going to divide 15 identical cookies among four people. How many ways are there to divide the cookies if all that matters is how many cookies a person receives? Redo this problem but now only consider divisions of the cookies where person $i$ gets at least $i$ cookies (thus person 1 must get at least one cookie, and so on). \#6: Redo the previous problem (15 identical cookies and 4 people), but with the following constraints: each person gets at most 10 cookies (it's thus possible some people get no cookies). \#7: Find a discrete random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for some $x$ between 17 and 17.01. \#8: Find a continuous random variable, or prove none exists, with probability density function $f_X$ such that $f_X(x) = 2$ for all $x$ between 17 and 17.01. \#9: Let $X$ be a continuous random variable with pdf $f_X$ satisfying $f_X(x) = f_X(-x)$. What can you deduce about $F_X$, the cdf? \#10: Find if you can, or say why you cannot, the first five Taylor coefficients of (a) $\log(1-u)$ at $u=0$; (b) $\log(1-u^2)$ at $u=0$; (c) $x \sin(1/x)$ at $x=0$. \#11: Let $X$ be a continuous random variable. (a) Prove $F_X$ is a non-decreasing function; this means $F_X(x) \le F_X(y)$ if $x < y$. (b) Let $U$ be a random variable with cdf $F_U(x) = 0$ if $u<0$, $F_U(x) = x$ if $0 < x < 1$, and $F_U(x) = 1$ if $1 < x$. Let $F$ be any continuous function such that $F$ is strictly increasing and the limit as $x$ approaches negative infinity of $F(x)$ is 0 and the limit as $x$ approaches positive infinity is 1. Prove $Y = F^{-1}(U)$ is a random variable with cdf $F$.

Week 4: September 24-28
- Videos: 2016: Mon: http://youtu.be/Jnqud_IcfAs Wed: http://youtu.be/8fYnAe7NZag Fri: http://youtu.be/dXBJ5F6pi5U
- Read: Chapter 5 and 6 of my textbook.
- Homework due Friday February 27: #1: Imagine we have a deck with s suits and N cards in each suit. We play the game Aces Up, except now we put down s cards on each turn. What is the probability that the final s cards are all in different suits? Write a computer program to simulate 1,000,000 deals and compare your observed probability with your theoretical prediction; it is fine to just do the program for s=4 and N=13 (a standard hand); you may earn 15 out of 10 points if you write general code for general s, N. #2: Consider all generalized games of Aces Up with C cards in s suits with N cards in a suit; thus C = sN. What values of s and N give us the greatest chance of all the cards being in different suits? Of being in the same suit? #3: The double factorial is defined as the product of every other integer down to 1 or 2; thus 6!! = 6 * 4 * 2 while 7!! = 7 * 5 * 3 * 1. One can write (2n-1)!! as a! / (b^c d!) where a, b, c and d depend on n; find this elegant formula! Hint: b turns out to be a constant, taking the same value for all n. #4: A regular straight is five cards (not necessarily in the same suit) of five consecutive numbers; aces may be high or low, but we are not allowed to wrap around. A kangaroo straight differs in that the cards now differ by 2 (for example, 4 6 8 10 Q). What is the probability someone is dealt a kangaroo straight in a hand of five cards? #5: A prisoner is given an interesting chance for parole. He's blindfolded and told to choose one of two bags; once he does, he is to reach in and pull out a marble. Each bag has 25 red and 25 black marbles, and the marbles all feel the same. If he pulls out a red marble he is set free; if it's a black, his parole is denied. What is his chance of winning parole? #6: The set-up is similar to the previous problem, except now the prisoner is free to distribute the marbles among the two bags however he wishes, so long as all the marbles are distributed. He's blindfolded again, chooses a bag at random again, and then a marble. What is the best probability he can do for being set free? While you can get some points for writing down the correct answer, to receive full credit you must prove your answer is optimal!

Week 3: September 17-21
- No written homework due next Friday! Instead use the time to build up your strategic reserve in the book. We will not cover most of Chapter 3 in class -- read the material and if there are calculations you are having trouble with or want to see in class, let me know and I'll do. Start reading Chapter Four. Monday's class will be a quick run of the Chapter Three material. Later we'll talk about some applications of probability to mathematical modeling. There will not be reading assigned for this; the purpose of this is to (1) quickly show you how useful probability can be, and (2) give you a sense of the tools and techniques we'll see later in the semester, and (3) give you plenty of time to read ahead and build up your strategic reserve (if you don't take advantage of this you'll have some painful weeks down the road!).
- Also use this time to make sure you can do simple, basic coding. I don't care what language you use (Mathematica, R, Python, Fortran, ...), but you should be comfortable doing simple assignments. I'll post a list of basic problems you should be able to do. If you want to learn Mathematica, I have a template online and a YouTube tutorial. Just go to http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm (note you'll also get links to using LaTeX). Here is a Mathematica file of code for problems in the book (with comments), as well as some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf). If you have trouble with any of these, please let me know -- you are responsible for making sure you can write code to solve at least half of these problems. There are also videos online from introducing coding in my problem solving class: 2017 here: https://youtu.be/e8I5alerkOw and 2018 here: https://youtu.be/e8I5alerkOw (code here: http://web.williams.edu/Mathematics/sjmiller/public_html/331Fa18/mathematicaprograms/math331IntroCoding2018.nb).

Week 2: September 11-14
- Read: Chapter 2 and Chapter 3 of my textbook
- Videos: Monday: http://youtu.be/dmI9d-w-bM4 Wed: http://youtu.be/Mg7xZqolBKE Fri: http://youtu.be/0Rp8KgWmLi0
- HW: Due September 14, 2018: #1: Section 1.2: Modify the basketball game so that there are 2015 players, numbered 1, 2, ..., 2015. Player i always gets a basket with probability 1/2ⁱ. What is the probability the first player wins? #2: Section 1.2: Is the answer for Example 1.2.1 consistent with what you would expect in the limit as c tends to minus infinity? #3: Section 1.2: Compute the first 42 terms of 1/998999 and comment on what you find; you may use a computer. #4: Section 2.2.1: Find sets A and B such that |A| = |B|, A is a subset of the real line and B is a subset of the plane (i.e., R²) but is not a subset of any line. #5: Section 2.2.1: Write at most a paragraph on the continuum hypothesis (you may use Wikipedia or any source to look it up). #6: Section 2.2.2: Give an example of an open set, a closed set, and a set that is neither open nor closed (you may not use the examples in the book); say a few words justifying your answer. #7: Section 2.3: Give another proof that the probability of the empty set is zero. #8: Find the probability of rolling exactly k sixes when we roll five fair die for k = 0, 1, ..., 5. Compare the work needed here to the complement approach in the book. #9: If f and g are differentiable functions, prove the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). Emphasize where you add zero.

Week 1: September 7, 2018
- Read: Chapter 1 of the textbook and the graduation speech here: http://www.graduationwisdom.com/speeches/0018-uslan.htm
- Homework: To be emailed to me by noon on Monday, September 10: Email me a short note on what you want to get out of this course, and what lesson you learned from the graduation speech. Full credit, 20/20, if you answer both questions on time. If you’ve already said what you want from a previous course with me, please briefly repeat. Thanks!
- click here for first day slides click here for first day's handout first day's video: http://youtu.be/pVQL4ivA08w