stat341 Homework 2 Autumn 2024Statistics

Java Python Homework 2

Autumn 2024

Instructions

• This homework is due in Gradescope on Wednesday Oct 23 by midnight PST. There is a 15 minute grace period and submissions made during this time will not be marked as late. Any work submitted past this period is considered late.

• Please answer the following questions in the order in which they are posed.

• Don’t forget to (i) make a local copy this document for your work and to (ii) knit the document frequently to make sure there are no compilation errors.

• When you are done, download the PDF file as instructed in section and submit it in Gradescope.

Exercises

1. (Joint PMF) The random variables X and Y are independent, each taking the values 1, 2 or 3. Complete the following table of the joint PMF. Show your work for each entry below the table.

2. (Two friends) Two friends - let’s call them Casey and Harry - agree to meet at Tully’s for coffee. Suppose that the random variables

X = the time that Casey arrives at Tully’s and

Y = the time that Harry arrives at Tully’s

are independent uniform. random variables on the interval [5, 6]. (The units are hours after noon)

a. Write the joint PDF f(x, y). Be sure to indicate the range of x and y.

b. Calculate the probability that both of them arrive before 5:30 PM.

Hint: You want to find P(X < 5.5, Y < 5.5)

c. Calculate the probability that Casey arrives before Harry. Show you stat341 Homework 2 Autumn 2024Statistics r steps. (Do the integration, do not just use geometry)

3. (Rounding errors) An individual makes 100 check transactions between receiving his December and his January bank statements. Rather than subtracting the amounts exactly, he rounds off each checkbook entry to the nearest dollar. Let Xi denote the round off error on the ith check. A reasonable assumption is that

independently of each other.

Let T = X1 + X2 + · · · + X100. The random variable T denotes the accumulated error after 100 transactions.

a. Find µ = E[T]. Show your steps.

b. Find σ = √Var[T]. Show your steps.

c. Using Chebychev’s inequality, what can you say about P(|T| ≥ 5)? Show your steps and write your answer in a complete sentence in context.

Hint: If you are unfamiliar, please consult the 394 Reading Guide for the statement of Chebychev’s inequality

4. (Gambling) A gambler plays n hands of poker. If she wins the k th hand she collects k dollars, while if she loses it, she collects nothing. Assume that her chances of winning each hand is constant (equal to p).

a. Let the random variable Xi denote her winnings on the ith hand. Write the PMF of Xi by filling in the empty cells in the table below.

b. Find µ = E[Xi ].

c. Let the random variable T denote the total amount she wins in n hands, that is T = X1 +X2 +· · ·+Xn. Find E[T]. Do we need to assume that the Xi to be independent for finding E[T]?

d. Find Var[T]. State assumptions (in context) you need to make about the Xi in order to do your calculation