University of Washington
Department of Mathematics Homework 4

MATH 394: Probability I

Instructor: Arman Jahangiri

Homework Policy

Submission and Deadline

Please:

• Submit homework as a single PDF on Gradescope by 10:00 PM (Pacific Time) on the listed due date.
• Begin each problem on a new page, and ensure that pages are correctly tagged in Gradescope.
• Write solutions clearly and legibly.
• Typesetting solutions in LaTeX is encouraged, but not required.

Late Days

Each student is allotted six late days for the quarter. A late day extends a homework deadline by up to 24 hours without penalty. For example, submitting an assignment anytime between 10:01 PM on the due date and 10:00 PM the following day counts as one late day.

The following rules apply:

• No explanation is required to use late days.
• At most two late days may be used on any single homework assignment.
• Late days may not be used on the final homework assignment or on any assignment for which solutions have already been released.
• It is the student’s responsibility to keep track of how many late days have been used during the quarter.

Once all late days have been exhausted, additional late submissions will incur a penalty of 10% per day, up to a maximum deduction of 50%. Assignments submitted more than five days late, or after solutions have been released, will not be accepted.

Submission Issues and Technical Difficulties

If a serious technical issue prevents a timely Gradescope submission, students may temporarily submit their assignment by email to the instructor at armanjg@uw.edu. In such cases:

• The submission must consist of a single PDF file. The email subject line must follow the format: \[ \texttt {HW\# Submission - FULL NAME - STUDENT ID}. \]
• Once the Gradescope issue has been resolved, the student must upload the exact same file to Gradescope as soon as possible.
• The student should clearly indicate in the Gradescope submission that the assignment was previously submitted by email before the deadline.

Email submissions are intended only for genuine technical emergencies and should not be used as a substitute for timely Gradescope submission.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 1.

People are arriving at a party one at a time. While waiting for more people to arrive, they entertain themselves by comparing their birthdays.

Let \(X\) be the number of people needed to obtain a birthday match; that is, before person \(X\) arrives, no two people have the same birthday, but when person \(X\) arrives there is a match.

Find the PMF of \(X\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 3.

Let \(X\) be a random variable with CDF \(F\), and let \[ Y=\mu +\sigma X, \] where \(\mu \) and \(\sigma \) are real numbers with \(\sigma >0\).

The random variable \(Y\) is called a location–scale transformation of \(X\).

Find the CDF of \(Y\) in terms of \(F\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 4.

Let \(n\) be a positive integer and define \[ F(x)=\frac {\lfloor x\rfloor }{n} \] for \(0\le x\le n\), where \(\lfloor x\rfloor \) is the greatest integer less than or equal to \(x\).

Also define \[ F(x)=0 \quad \text {for } x<0, \] and \[ F(x)=1 \quad \text {for } x>n. \]

(a)

Show that \(F\) is a valid CDF.

(b)

Find the PMF corresponding to \(F\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 5.

(a)

Show that \[ p(n)=\left (\frac 12\right )^{n+1}, \qquad n=0,1,2,\dots , \] is a valid PMF for a discrete random variable.

(b)

Find the CDF of a random variable with the PMF from part (a).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 8.

There are 100 prizes, with one worth $1, one worth $2, …, and one worth $100. There are 100 boxes, each containing one of the prizes.

You get 5 prizes by picking random boxes one at a time, without replacement.

Find the PMF of how much your most valuable prize is worth, as a simple expression in terms of binomial coefficients.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 9.

Let \(F_1\) and \(F_2\) be CDFs, let \(0<p<1\), and define \[ F(x)=pF_1(x)+(1-p)F_2(x) \] for all \(x\).

(a)

Show directly that \(F\) has the properties of a valid CDF. The distribution defined by \(F\) is called a mixture of the distributions defined by \(F_1\) and \(F_2\).

(b)

Consider creating a random variable in the following way. Flip a coin with probability \(p\) of Heads. If the coin lands Heads, generate a random variable according to \(F_1\); if the coin lands Tails, generate a random variable according to \(F_2\). Show that the random variable obtained in this way has CDF \(F\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 22.

There are two coins, one with probability \(p_1\) of Heads and the other with probability \(p_2\) of Heads. One of the coins is randomly chosen, with equal probabilities for the two coins. It is then flipped \(n\ge 2\) times.

Let \(X\) be the number of times it lands Heads.

(a)

Find the PMF of \(X\).

(b)

What is the distribution of \(X\) if \(p_1=p_2\)?

(c)

Give an intuitive explanation of why \(X\) is not Binomial for \(p_1\ne p_2\). You may assume that \(n\) is large for your explanation.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 3, Problem 30.

A certain company has \(n+m\) employees, consisting of \(n\) women and \(m\) men. The company is deciding which employees to promote.

(a)

Suppose for this part that the company decides to promote \(t\) employees, where \[ 1\le t\le n+m, \] by choosing \(t\) random employees, with equal probabilities for each set of \(t\) employees.

What is the distribution of the number of women who get promoted?

(b)

Now suppose that instead of having a predetermined number of promotions to give, the company decides independently for each employee, promoting the employee with probability \(p\).

Find the distributions of:

• the number of women who are promoted,
• the number of women who are not promoted,
• the number of employees who are promoted.

(c)

In the set-up from part (b), find the conditional distribution of the number of women who are promoted, given that exactly \(t\) employees are promoted.