Probability: Homework 6

pdfs and cdfs

1. Show that $f(x) = x e^{-x^2 / 2}$ ($x > 0$) is a valid pdf. A random variable with this density is called a Rayleigh random variable.
2. Find $P(1 < X < 3)$ if $X$ is a Rayleigh variable.

Let $U \sim \text{Uniform}(0, 8)$.
1. Find $P(U \in (0, 2) \cup (3, 7))$ without any calculus.
2. Compute $P(U \leq x \mid U \in (3, 7))$. This is called the conditional distribution of $U$ given $3 \leq U \leq 7$. What named distribution is it?
Let $U$ be uniform on $(-1, 1)$.
1. Compute $E[U^n]$ for $n \geq 0$.
2. Compute $\sum_{n = 0}^\infty \frac{E[U^n]}{n!} x^n$. (This is the moment generating function of $U$. We’ll come back to this next week.)
3. Find the pdf of $U^2$. Is $U^2$ uniform on $(0, 1)$?
A stick of length 1 is broken at a uniformly random point. Find the cdf and expected value of the length of the longer piece.
A stick of length 1 is broken at a uniformly random point. Let $X$ be the length of the longest part, and $Y$ the length of the shortest part. Find the cdf and pdf of $X / Y$.

Hint. $X = \max(U, 1 - U)$ where $U$ is uniform(0, 1). What’s the relationship between $X$ and $Y$?
Let $U_1, U_2, \dots, U_n$ be independent uniform(0, 1) variables. Find the CDF of $X = \max(U_1, U_2, \dots, U_n)$, the largest of the $n$ variables.

Hint. $X \leq x$ if and only if $U_1 \leq x, U_2 \leq x, \dots, U_n \leq x$.
Let $U \sim \text{Uniform}(0, 1)$. Show that $Y = -\lambda^{-1} \ln(1 - U)$ is an exponential random variable with parameter $\lambda$.

Hint. Two random variables have the same distribution if they have the same cdf.

The time $T$ that a radioactive particle takes to decay is well-modeled by an exponential variable, say $T \sim \text{Exp}(\lambda)$.

The half-life of a particle is the time at which there is a 50% chance the particle has decayed. Find the half-life.
A post office has two clerks. Alice enters the post office while two other customers, Bob and Claire, are being served by two clerks. Assume that the time a clerk spends serving a customer has an Exp($\lambda)$ distribution.
1. What is the probability that Alice is the last of the three customers to be done being served?
2. What is the expected total time that Alice needs to spend at the post office?
You’re selling a CS2 knife and won’t accept less than $500. (If you’re a normal person, change ‘CS2 knife’ into ‘completely useless but expensive object.’) Suppose that the bids you get are independent Exponential(450) variables.
1. Let $X$ be the number of offers it takes before you sell. What is the distribution of $X$? What is $E[X]$?
2. What is the expected amount of money you will gain from the sale? Hint. Use the memoryless property of the exponential.
You call two ride-sharing services at the same time. The number of minutes until they arrive at your location follow Exp(6) and Exp(7) distributions, respectively. You will get in the first car that arrives.
1. What is the probability that the Exp(6) takes more than 6 minutes to arrive?
2. What is the expected number of minutes you will wait to leave?

Alice is trying to transmit to Bob the answer to a yes-no question, using a noisy channel. She encodes “yes” as 1 and “no” as 0, and sends the appropriate value. However, the channel adds noise; specifically, Bob receives what Alice sends plus a $N(0, \sigma^2)$ noise term (the noise is independent of what Alice sends). If Bob receives a value greater than 1/2 he interprets it as “yes”; otherwise, he interprets it as “no”.

Find the probability that Bob understands Alice correctly.
What happens to the result from (a) if $\sigma$ is very small? What about if $\sigma$ is very large? Explain intuitively why the results in these extreme cases make sense.

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 can’t measure the distance exactly. Instead, we will observe a value of 10+$\epsilon$, where the error is normally distributed with mean $0$ and standard deviation 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 the standard normal cdf $\Phi$ and an approximate numerical answer (using a computer).
A woman is pregnant, with a due date of January 10, 2020. Of course, the actual date on which she will give birth is not necessarily the due date. On a timeline, define time 0 to be the instant when January 10, 2020 begins. Suppose that the time T when the woman gives birth has a Normal distribution, centered at 0 and with standard deviation 8 days. What is the probability that she gives birth on her due date? (Your answer should be in terms of the standard normal cdf, and simplified.)