Probability

Due date: Oct 4, 2021 at 11:59pm You are encouraged to talk with each other, but the work you submit must be your own. You only need to submit your solutions to Problems 1–6, but you should attempt all other problems as well. There will be a quiz on Oct 8, 2021 covering similar (but not necessarily identical) problems.

Problem 1: (2+2=4 points) Suppose that X and Y have the following joint distribution P(X = x,Y = y), where c and d are parameters.

Y = 0 Y = 1 ∑

X = −1 c 14 − c 1 4

X = 0 d 12 −d 1 2

X = 1 12 − c−d c + d− 1 4

1 4∑ 1

2 1 2 1

a) Give all valid values for c and d (Hint: Use equations such as c ≥ 0 to describe the set of valid pairs (c,d) or use a graphical argument).

b) For which values of c and d are X and Y independent?

Problem 2 (p.160, #15): (1+1+1+2+2+2=9 points) Let X and Y be independent, each uniformly distributed on {1, 2, . . . ,n}. Find:

a) P(X = Y ); b) P(X < Y ); c) P(X > Y ); d) P(max(X,Y ) = k) for 1 ≤ k ≤ n; e) P(min(X,Y ) = k) for 1 ≤ k ≤ n; f) P(X + Y = k) for 2 ≤ k ≤ 2n [Hint: Consider two cases: 2 ≤ k ≤ n + 1 and n + 2 ≤ k ≤ 2n].

Problem 3 (p.161, #24(a)): (2 points) Suppose a box contains tickets, each labeled by an integer. Let X and Y be the results of draws at random with replacement from the box: Show that, no matter what the distribution of numbers in the box,

P(X + Y is even) ≥ 1 2 .

Problem 4: (2 points) a) You want to invent a gambling game in which a person rolls two dice and is paid some money if

the sum is 7, but otherwise lose their money. How much should you pay for winning a $1 bet if you want this to be a fair game, that is, for the amount won by the gambler to have expectation 0?

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b) In another gambling game, three dice are rolled. For a bet of $1 you win $1 for each 6 that appears (plus your original dollar back). If no 6 appears then you lose your dollar. What is the expected value of your winnings from playing this game once?

Problem 5 (p.182, #4): (1 point) Suppose all the numbers in a list of 100 numbers are non-negative, and that the average of the numbers in the list is 2. Show that at most 25 of the numbers in the list are greater than 8.

Problem 6 (p.183 #14): (2 points) A building has 10 floors above the basement. If 12 people get into an elevator at the basement, and each chooses a floor at random to get out, independently of the others, how many floors do you expect the elevator to make a stop to let out one ore more of these 12 people? [Hint: Use the methods of indicators].

Additional problems that will not be graded and do not need to be submitted:

Problem 7 (p.158, #6): A fair coin is tossed three times. Let X be the number of heads on the first two tosses, and let Y be the number of heads on the last two tosses.

a) Make a table showing the joint distribution of X and Y . b) Are X and Y independent? c) Find the distribution of X + Y .

Problem 8 (p.159, #7): Let A,B, and C be events that are independent, with probabilities a,b, and c. Let N be the random number of events that occur.

a) Express the event (N = 2) in terms of A,B, and C. b) Find P(N = 2).

Problem 9: Suppose X ∼ Geometric(p) for some p ∈ [0, 1] (i.e. the distribution of X is the Geometric(p) distri- bution). Recall that this means P(X = k) = (1 −p)k−1p for k = 1, 2, 3, . . ..

a) Show that P(X > n) = (1 −p)n for every n ≥ 1. b) Show that for every n ≥ 1,

P(X = n + k|X > n) = P(X = k) for k ≥ 1.

[This is called the memoryless property: it says that the number of additional trials until the first success given that there has been no success through the nth trial has the same distribution as the (unconditional) number of trials until the first success: it is as if we have forgotten that the first n trials happened.]

Problem 10 (p.160, #16): Discrete convolution formula Let X and Y be independent random variables with non-negative integer values. Show that:

a) P(X + Y = n) = ∑n

k=0 P(X = k)P(Y = n−k).

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b) Find the probability that the sum of numbers on four dice is 8, by taking X to be the sum on two of the dice, Y the sum on the other two.

Problem 11 (p.182, #10): Let A and B be independent events, with indicator random variables IA and IB .

a) Describe the distribution of the random variable (IA + IB )2 in terms of P(A) and P(B). b) What is E[(IA + IB )2]?

Problem 12 (p.183, #18): Suppose X is a random variable with just two possible values a and b. For x = a and x = b, find a formula for p(x) = P(X = x) in terms of a,b, and µ = E(X)