INSTRUCTIONS

Solve the following problems and upload a PDF containing your work to Gradescope before the start of class on Tuesday 6/20. You are allowed to work with any of your classmates and use any resources found in your textbook or online, as long as your writeup is your own. You will receive full credit for correctness as well as showing all your work (do not just put down the final answer to a problem unless this is explicitly asked for).  Even if you do not ultimately have the correct answer, showing your work will yield you significant partial credit. If any question is labeled with a (P), it will solely be graded on a complete attempt rather than correctness. As always, feel free to ask me any questions that arise!

Problem 1

Compute the joint PMFs/PDFs for each of the following pairs of random variables (if the pairs happen to be discrete, also include the marginal PMFs):

(a) X ∼ Bernoulli(p), Y fair 6-sided dice roll.

(b) X ∼ N(µ1 ,σ1(2)), Y ∼ N(µ2 ,σ2(2)) (assume that X,Y are independent).

(c) X ∼ Exp(λ), Y ∼ Unif([0, 1]).

For the discrete pairs, you should present your joint and marginal PMFs as tables.  For continuous, you should write the PDFs as functions over R.

Problem 2 (P)

In class, we demonstrated that the joint PMF of two discrete random variables satisfied the exhaustion property, namely对x 对y pX,Y (x,y) = 1. Now consider the collection of random variables {Xk }k(n)=1 , for some n ∈ N.

(a) Write out in words why you believe that the joint PMF pX1,X2 ,...,Xn(x1 ,x2 , . . . ,xn )

satisfies the exhaustive property.

(b) Prove mathematically the claim in part (a):

工工 ··· 工 pX1,X2 ,...,Xn (x1 ,x2 , . . . ,xn ) = 1,    for all n ∈ N

x1       x2                 xn

You can prove this either constructively or with a technique like induction – up to you!

Problem 3

Consider Sn  = 对k(n)=1 Xk  for a collection of random variables {Xk }k(n)=1 .  Prove for arbitrary n that E(Sn ) = 对k(n)=1 E(Xk ) for both (a) discrete and (b) continuous collections of random variables. Hint : generalize our proof in class for a sum of two random variables

Problem 4

Let {Xn }n(N)=1 be a collection of i.i.d. Bernoulli random variables with success parameter equal to p, where N ∼ Poisson(λ) is a random parameter.  Let SN  = 对n(N)=1 Xn . What is E(SN )? Hint:  do not try to apply linearity of expectation – think about the significance of N being a random parameter and how the expectation of SN  changes depending on the value of N .

Problem 5

In this problem we will study the game of roulette, a popular and seemingly fair gambling game, relative to general casino standards of course.  If you unfamiliar with the game of roulette, here is a brief summary of the rules (adapted for our problem):

• Every turn the gambler randomly drops a ball into a spinning wheel with slots consisting of integers 1-36, and a certain number of spaces marked with 0.

• Half (18) of the nonzero spaces are colored red, the other half of the nonzero spaces are colored black, and spaces with 0 are colored green.

• Each turn you can make one of three bets (assuming you bet $1):

1) Bet on a single nonzero number with payout = $36

2) Bet on either red or black, with payout = $2

3) Bet on the first third of the nonzero integers (1-12), the second third (13-24), or the last third (25-36), with payout = $3

If the ball lands in a spot in which you have placed a bet, you win your respective payout. If not, then you lose. Assume landing on each space is equally likely.

We will study several outcomes of the game of roulette. In our analysis below we will refer to the three above betting strategies in the order in which they are listed:

(a) Suppose the roulette wheel has no spaces with 0. Compute the expected payout of each of the three strategies.

(b) Now suppose the roulette wheel adds exactly one space with 0.  Compute the new

expected payout of each of the three strategies.

(c) Compute the expected payout of each of the three strategies for any general z  ∈ N number of spaces with a 0.

(d) Suppose you are feeling lucky one night and decide to randomly choose a strategy. Let p1  be the probability that you choose strategy 1, p2  be the probability that you choose strategy 2, and p3  be the probability that you choose strategy 3. Assuming that there are a nonzero z ∈ N number of spaces with the value 0, what are your expected net winnings? (Note: net winnings means “payout − amount bet”)

(e) Given your result in the previous part, justify why you should or should not choose to

play roulette. Hint: think about how the exact value of z affects your net winnings.

Problem 6

Recall problem 8 from problem set 1. In this problem, we used the law of total probability in order to determine the likelihood of the third hitter in a softball batting lineup getting a hit. We can think of each of the three hitters’ hit outcomes (1 = hit, 0 = out) as a discrete random variable:  let Bk  be a binary random variable that represents whether or not the kth  batter gets a hit.  Letting H  = 对k(3)=1 Bk , we will compute E(H) using the following procedure:

(a) Using the rules defined in problem set 1, compute the joint PMF pB1,B2 ,B3 (b1 ,b2 ,b3 ).

For reference,

1)   P(B1  = 1) = .300

2)   P(B2  = 1 | B1  = 0) = .280,    P(B2  = 1 | B1  = 1) = .200

3)   P(B3  = 1 | B1  = 0,B2  = 0) = .250,    P(B3  = 1 | B1  = 1,B2  = 1) = .350,

P(B3  = 1 | B1  = 1,B2  = 0) = (B3  = 1 | B1  = 0,B2  = 1) = .300,

(b) Are B1 , B2 , and B3  independent? Why or why not?

(c) Use this joint PMF in order to verify your answer from problem set 1, namely, compute the marginal PMF of B3 .

(d) Rewrite E(H) in terms of E(B1 ), E(B2 ), and E(B3 ).

(e) Without computing E(B1 ), E(B2 ), or E(B3 ), rank what you believe will be the order

from lowest to highest and explain why you chose what you did.  I am not looking necessarily for the correct ranking, but rather just to hear your intuition.

(f) Compute E(B1 ), E(B2 ), and E(B3 ), and based off your answer in part (d), determine

what is E(H).

For problems 7 and 8, you will need the attached excel spreadsheet

Problem 7

In this problem we will take a numerical approach to the previous problem. You should use the sheet labeled “Problem 7”for this question.  Note:  you will need the answer to problem 6 for this question.  If you cannot solve that problem, use a value you think makes sense and just comment that you used a random value. I will grade solely off the accuracy of the methods used in this question alone.

(a) First turn your attention to the table at the top of the sheet. Here I have entered all

the statistics that we used in problem 6.  Why is it important that we make note of how the averages change relative to the number of runners on base (AKA people that got a hit)?

(b) Click on cell B11. Carefully observe the formula I have written in there. Try to write

out the formula using words. For example, if the formula was “=AVERAGE(B5:B7)”, you would write “this is the average value of the data entered in cells B5 through B7.”

(c) Enter your answer (or, if you have none, a random reasonable answer) from problem

6 into cell B17.  Turn your attention to the percent error between the simulation and your answer in cell B18. After running the random number generator a few times, how accurate does the simulation appear to be?

(d) We can represent the sum of each column as a random variable Hk , and therefore the value in cell B16 as

对k(n)=1 Hk n ,    n = 100.

Based on what we have learned in class this week, what theorem are we attempting to apply in order to approximate the answer in problem 6?

(e) Extend this experiment out to 325 trials.  That is, extend rows 8 through 12 out to

column LN (I have shaded all the relevant cells yellow). Make sure to change cell B14 to the number 325 as well. How does this affect our percent error in cell B18? Based on your answer in part (d), why does extending the number of trials make our percent error better/worse/the same?