Homework 5

Problem 1

(Continuing from a previous HW) The latest series of collectible Lego Minifigures contains 3 different Minifigure prizes (labeled 1, 2, 3). Each package contains a single unknown prize. Suppose we only buy 3 packages and we consider as our sample space outcome the results of just these 3 packages (prize in package 1, prize in package 2, prize in package 3). For example, 323 (or (3, 2, 3)) represents prize 3 in the first package, prize 2 in the second package, prize 3 in the third package. Let \(X\) be the number of distinct prizes obtained in these 3 packages. Let \(Y\) be the number of these 3 packages that contain prize 1. Suppose that each package is equally likely to contain any of the 3 prizes, regardless of the contents of other packages; let \(\text{P}\) denote the corresponding probability measure.

Find the conditional distribution of \(Y\) given \(X=x\) for each possible value of \(x\) of \(X\).
Compute and interpret \(\text{E}(Y|X=x)\) for each possible value of \(x\) of \(X\).
Find the conditional distribution of \(X\) given \(Y=y\) for each possible value of \(y\) of \(Y\).
Compute and interpret \(\text{E}(X|Y=y)\) for each possible value of \(y\) of \(Y\).
Explain how you could use spinners to implement the “marginal then conditional” method to simulate an \((X, Y)\) pair.
Suppose you have simulated many \((X, Y)\) pairs. Explain how you could use the simulation results to approximate:
1. \(\text{P}(X = 1 | Y = 0)\)
2. the conditional distribution of \(X\) given \(Y=0\)
3. \(\text{E}(X| Y = 0)\)

Problem 2

Xavier and Yolanda are playing roulette. They both place bets on red on the same 3 spins of the roulette wheel before Xavier has to leave. (Remember, the probability that any bet on red on a single spin wins is 18/38.) After Xavier leaves, Yolanda places bets on red on 2 more spins of the wheel. Let \(X\) be the number of bets that Xavier wins and let \(Y\) be the number that Yolanda wins.

Identify by name the marginal distribution of \(X\). Be sure to specify the values of any relevant parameters. Compute \(\text{E}(X)\).
Identify by name the marginal distribution of \(Y\). Be sure to specify the values of any relevant parameters. Compute \(\text{E}(Y)\).
The joint distribution of \(X\) and \(Y\) is represented in the table below. Explain why \(p_{X, Y}(1, 4) = 0\) and \(p_{X, Y}(2, 1) = 0\).
Compute \(p_{X, Y}(2, 3)\). (Yes, the table tells you it’s 0.1766, but you have to show how this number can be computed based on the assumptions of the problem.)
Are \(X\) and \(Y\) independent? Justify your answer with an appropriate calculation.
Compute and interpret \(\text{P}(X + Y) = 4\).
Make a table representing the marginal distribution of \(X+Y\).
Use the table from the previous part to compute \(\text{E}(X+Y)\), and verify that it is equal to \(\text{E}(X)+\text{E}(Y)\).
Without doing any computation, determine if \(\text{Var}(X+Y)\) will be greater than, less than, or equal to \(\text{Var}(X) + \text{Var}(Y)\). Explain your reasoning.

\(x\), \(y\)	0	1	2	3	4	5
0	0.0404	0.0727	0.0327	0	0	0
1	0	0.1090	0.1963	0.0883	0	0
2	0	0	0.0981	0.1766	0.0795	0
3	0	0	0	0.0294	0.0530	0.0238

Problem 3

Percent returns for assets \(X\), \(Y\), and \(Z\) follow a joint distribution with

Mean 10 and standard deviation 15 for asset X,
Mean 5 and standard deviation 3 for asset Y,
Correlation of −0.6 between asset X and asset Y
Asset Z yields a constant return of 1 percent.

An investment portfolio has 60% of its funds in asset X, 30% in asset Y, and 10% in asset Z.

Let \(R\) be the portfolio return. Express \(R\) in terms of \(X, Y, Z\).
Compute \(\text{E}(R)\).
Compute \(\text{Cov}(X, Y)\).
Compute \(\text{Cov}(X, Z)\).
Compute \(\text{SD}(R)\).

Problem 4

Devi and Paxton are meeting. Arrival times are measured in minutes after noon, with negative times representing arrivals before noon. Devi’s arrival time follows a Normal distribution with mean 20 and SD 15 minutes, and Paxton’s arrival time follows a Normal distribution with mean 25 and SD 10 minutes, independently of each other.

For each of the following, find the appropriate standardized value and make a ballpark estimate of the probability. Then use software to compute the probability.

Compute the probability that Devi arrives before noon.
Compute the probability that Devi arrives first.
Compute the probability that the first person to arrive has to wait more than 15 minutes for the second person to arrive.
Coding required. Code and run a simulation and use the results to approximate the probability from the previous part (just part c). You should first simulate (Devi, Paxton) arrival time pairs, and go from there.

Problem 5

(Continued.) Devi and Paxton are meeting. Arrival times are measured in minutes after noon, with negative times representing arrivals before noon. Devi’s arrival time follows a Normal distribution with mean 20 and SD 15 minutes, and Paxton’s arrival time follows a Normal distribution with mean 25 and SD 10 minutes.

For each of the following, find the appropriate standardized value and make a ballpark estimate of the probability. Then use software to compute the probability.

Assume the pairs of arrival times follow a Bivariate Normal distribution with correlation 0.8
1. Compute the probability that Devi arrives first given that Paxton arrives at 12:10.
2. Compute the probability that the first person to arrive has to wait more than 15 minutes for the second person to arrive.
3. This part is optional. Code and run a simulation and use the results to approximate the probability from the previous part (just part b). You should first simulate (Devi, Paxton) arrival time pairs, and go from there.
Assume the pairs of arrival times follow a Bivariate Normal distribution with correlation -0.7
1. Compute the probability that Devi arrives first given that Paxton arrives at 12:10.
2. Compute the probability that the first person to arrive has to wait more than 15 minutes for the second person to arrive.
3. This part is optional. Code and run a simulation and use the results to approximate the probability from the previous part (just part b). You should first simulate (Devi, Paxton) arrival time pairs, and go from there.