University of Washington
Department of Mathematics Homework 6

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 5, Problem 4.

Let \(X\) be a continuous random variable with CDF \(F\) and PDF \(f\).

(a)

Find the conditional CDF of \(X\) given \(X>c\), for a constant \(c\) with \(P(X>c)\ne 0\). That is, find \(P(X\le x\mid X>c)\) for all \(x\), in terms of \(F\).

(b)

Find the conditional PDF of \(X\) given \(X>c\).

(c)

Check that the conditional PDF from part (b) is a valid PDF, by showing directly that it is nonnegative and integrates to 1.

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

A circle with a random radius \(R\sim \operatorname {Unif}(0,1)\) is generated. Let \(A\) be its area.

(a)

Find the mean and variance of \(A\), without first finding the CDF or PDF of \(A\).

(b)

Find the CDF and PDF of \(A\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 12.

A stick is broken into two pieces, at a uniformly random breakpoint. Find the CDF and average of the length of the longer piece.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 14.

Let \(U_1,\dots ,U_n\) be i.i.d. \(\operatorname {Unif}(0,1)\), and \[ X=\max (U_1,\dots ,U_n). \] What is the PDF of \(X\)? What is \(E(X)\)?

Hint: Find the CDF of \(X\) first, by translating the event \(X\le x\) into an event involving \(U_1,\dots ,U_n\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 15.

Let \(U\sim \operatorname {Unif}(0,1)\). Using \(U\), construct \[ X\sim \operatorname {Expo}(\lambda ). \]

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 22.

The distance between two points needs to be measured, in meters. The true distance between the points is 10 meters, but due to measurement error we cannot measure the distance exactly. Instead, we will observe a value of \(10+\epsilon \), where the error \(\epsilon \) is distributed \[ N(0,0.04). \]

Find the probability that the observed distance is within \(0.4\) meters of the true distance, 10 meters. Give both an exact answer in terms of \(\Phi \) and an approximate numerical answer.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 29.

Let \[ Y=|X|, \qquad X\sim N(\mu ,\sigma ^2). \] This is a well-defined continuous random variable, even though the absolute value function is not differentiable at 0.

(a)

Find the CDF of \(Y\) in terms of \(\Phi \). Be sure to specify the CDF everywhere.

(b)

Find the PDF of \(Y\).

(c)

Is the PDF of \(Y\) continuous at 0? If not, is this a problem as far as using the PDF to find probabilities?

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 36.

A post office has 2 clerks. Alice enters the post office while 2 other customers, Bob and Claire, are being served by the 2 clerks. She is next in line. Assume that the time a clerk spends serving a customer has an \(\operatorname {Expo}(\lambda )\) distribution.

(a)

What is the probability that Alice is the last of the 3 customers to be done being served?

Hint: No integrals are needed.

(b)

What is the expected total time that Alice needs to spend at the post office?

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 5, Problem 43.

The Exponential is the analog of the Geometric in continuous time. This problem explores the connection between Exponential and Geometric in more detail, asking what happens to a Geometric in a limit where the Bernoulli trials are performed faster and faster but with smaller and smaller success probabilities.

Suppose that Bernoulli trials are being performed in continuous time; rather than only thinking about first trial, second trial, etc., imagine that the trials take place at points on a timeline. Assume that the trials are at regularly spaced times \[ 0,\Delta t,2\Delta t,\dots , \] where \(\Delta t\) is a small positive number. Let the probability of success of each trial be \(\lambda \Delta t\), where \(\lambda \) is a positive constant. Let \(G\) be the number of failures before the first success in discrete time, and let \(T\) be the time of the first success in continuous time.

(a)

Find a simple equation relating \(G\) to \(T\).

Hint: Draw a timeline and try out a simple example.

(b)

Find the CDF of \(T\).

Hint: First find \(P(T>t)\).

(c)

Show that as \(\Delta t\to 0\), the CDF of \(T\) converges to the \(\operatorname {Expo}(\lambda )\) CDF, evaluating all the CDFs at a fixed \(t\ge 0\).

Hint: Use the compound interest limit.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 1–2.

(a)

Let \(U\sim \operatorname {Unif}(a,b)\). Find the median and mode of \(U\).

(b)

Let \(X\sim \operatorname {Expo}(\lambda )\). Find the median and mode of \(X\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 13.

A fair die is rolled twice, with outcomes \(X\) for the first roll and \(Y\) for the second roll. Find the moment generating function \[ M_{X+Y}(t) \] of \(X+Y\). Your answer should be a function of \(t\) and may contain unsimplified finite sums.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 15.

Let \[ W=X^2+Y^2, \] with \(X,Y\) i.i.d. \(N(0,1)\). The MGF of \(X^2\) turns out to be \[ (1-2t)^{-1/2},\qquad t<\frac 12, \] and you may assume this.

(a)

Find the MGF of \(W\).

(b)

What famous distribution that we have studied so far does \(W\) follow? Be sure to state the parameters in addition to the name.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 17.

Let \(X_1,\dots ,X_n\) be i.i.d. with mean \(\mu \), variance \(\sigma ^2\), and MGF \(M\). Let \[ Z_n=\sqrt n\left (\frac {\bar X_n-\mu }{\sigma }\right ). \]

(a)

Show that \(Z_n\) is a standardized quantity; that is, it has mean 0 and variance 1.

(b)

Find the MGF of \(Z_n\) in terms of \(M\), the MGF of each \(X_j\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 21.

Let \[ X_n\sim \operatorname {Bin}(n,p_n) \] for all \(n\ge 1\), where \(np_n\) is a constant \(\lambda >0\) for all \(n\), so that \[ p_n=\frac {\lambda }{n}. \] Let \[ X\sim \operatorname {Pois}(\lambda ). \]

Show that the MGF of \(X_n\) converges to the MGF of \(X\). This gives another way to see that the \(\operatorname {Bin}(n,p)\) distribution can be well-approximated by the \(\operatorname {Pois}(\lambda )\) distribution when \(n\) is large, \(p\) is small, and \(\lambda =np\) is moderate.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 6, Problem 23.

Let \(U_1,U_2\) be i.i.d. \(\operatorname {Unif}(0,1)\). The random variable \(U_1+U_2\) has a Triangular distribution, with PDF \[ f(t)= \begin {cases} t, & 0<t\le 1,\\ 2-t, & 1<t<2. \end {cases} \]

Show that \(U_1+U_2\) has a Triangular distribution by showing that they have the same MGF.