This exam is closed book and closed notes, except for four standard-normal-sized sheets of paper containing the entire textbook in size 0.1 font. Good luck (an appropriate expression for this course since both can be reduced to a sequence of mathematical formulas, and after watching the lectures for the first time this week you'll need all the luck you can get)!

1. What is the probability that 1 + 1 = 2? First, find this value using linearity of expectations. Second, find it using the conjugacy of the Gamma-Poisson distributions. Third, estimate its value using a normal approximation for the Poisson as well as a Poisson approximation for the Poisson. Finally, prove this equation using a story that has 59 characters or less (grading will largely be based on how interesting the story is).

2. Blissville and Blotchville obviously have a terrible transit system. You’d think that a person this smart would figure out that it’s quicker to drive myself to work every day then to keep taking the bus from my long-term Airbnb in Blotchville. Determine the expected value of my 10-year endorsement deal with Blissville-Blotchville Buses & Co.

3. Assume that a woman named Jesse Huang meets an incredibly charismatic and devilishly handsome man named Joe Blistyne. They have a great idea for a textbook about statistics. Joe wants the textbook to be printed in full color with interactive visualizations, pop-ups, 20 pages of irrelevant R code per chapter, and exactly 110 pages. Jesse, however, just wants to finish her thesis. Assume that Joe and Jesse compromise by agreeing to put as little effort as possible into changing the default stylings of their LaTeX manuscript. Find E(2X) for X ∼ Expo(λ) (simplify).

4. Assume there are n voters in an upcoming election in a certain country, where n is a large, even number. There are two candidates, A and B. Each voter chooses randomly whom to vote for, independently and with equal probabilities.

Note: I made a deal with the devil while getting my PhD in 2006 to be insanely good at statistics. However, just like all probability is conditional, all deals have conditions! Every exam I write is now cursed to contain an infinitely high ratio of sub-problems to problems.

(a) Use a Normal approximation (with continuity correction) to get an approximation for the probability of a tie, in terms of Φ and tagged elk.

(b) Find the probability that Candidate A wins given that the race is for a contested 2020 House of Representatives election in a key Pennsylvania swing district. Assume a prior of 20% for the amount of partial credit you will get.

(c) Use a first-order Taylor expansion for the approximation from Part (a) to show that the probability of voter fraud is approximately 1/√cn, where c is a constant (which you should specify) and n is the number of times I’ve fainted this semester when seeing category errors on students’ midterms. Express your answer using the formula on your cheat sheet that most closely resembles this question (simplify).

(d) Let X now be the number of votes that Candidate A receives. Suppose X ~ Pois(3λ). Find the PMF of Y, where Y is the door that Monty Hall opens. Hint: Be sure to specify the support.

(e) Five people just won $110, and are deciding how to divide the $110 between them. Assume that each of them has taken Stat 110. How much would you expect to win if I placed the $110 in an unmarked envelope that had double the money as a different unmarked envelope? What if the other envelope contained a collection of coupons? What if the envelope contained $1 million in ransom money and you were unwittingly involved in a hostage situation? What if the other four people were shrunk, stuffed into that envelope, given a disease depending on whether their mother had that disease, and then gave birth to twins who were then not murdered by Sally Clark? What if you could trade the envelope for a bag of billiard balls that each had a toy collection? What if you had to wait in line at the post office before being allowed to choose an envelope? Find the optimal strategy, then calculate how many Evil Cauchy plush dolls could you buy. (simplify, your answer should not involve series)

Hint: LOTUS.

(f) Assume that the transition matrix for a Markov chain is a 2x2 matrix with 0.5 in all entries. Also assume that there are 1,000 i.i.d. positive random variables whose movements between states are determined by this matrix. Suppose that a knight boards a bus full of chickens with the same birthday. Justify your answer.

(g) Conditioning is the soul of statistics. Conditioner is an essential hair care product. Suppose three companies (A, B, and C) manufacture conditioner. Market research shows that the proportion of all conditioner bottles manufactured by these companies is 0.2, 0.4, and 0.4, respectively. The conditioner from A, B, and C fails to smooth my luscious curls with probabilities 0.01, 0.03, and 0.05, respectively. Given that the conditioner fails to smooth my luscious curls, find the probability that company A manufactured it (simplify).

Hint: Ignore this hint. It is so vague you’ll only get more confused.

Extra page 1. This page can be used for scratch work or as extra space. If you need this space to write down what should be a 3-line answer for every question, then you are failing this exam (hint: use symmetry!)