Homework #6

EE 364: Spring 2026

Assignment Details

Assigned: 25 February
Due: Thursday, 05 March at 16:00

BrightSpace Assignment: Homework 6

Instructions

Write your solutions to these homework problems. Submit your work to BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.

Problem 0

Daily derivation #6, Prove $\Gamma(\alpha + 1) = \alpha \, \Gamma(\alpha)$ for $\alpha > 0$. Then show $\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}$.

Problem 1

A function $X: \Omega \to \mathbb{R}$ is a random variable on $(\Omega, \mathcal{A})$ if the pullback $X^{-1}(B) \in \mathcal{A}$ for every Borel set $B \in \mathcal{B}(\mathbb{R})$, that is PAM — pullbacks are always measurable. Suppose $\Omega = \{1, 2, 3, 4, 5, 6\}$ with $\sigma$-algebra $\mathcal{A} = \{\emptyset, \{1,2,3\}, \{4,5,6\}, \Omega\}$. Define $X: \Omega \to \mathbb{R}$ by $X(\omega) = (\omega - 3)^2$. Is $X$ a random variable on $(\Omega, \mathcal{A})$?

Problem 2

The indicator function $\mathbb{1}_A: \Omega \to \{0, 1\}$ for a subset $A \subseteq \Omega$ assigns $\mathbb{1}_A(\omega) = 1$ if $\omega \in A$ and $\mathbb{1}_A(\omega) = 0$ if $\omega \notin A$. Let $(\Omega, \mathcal{A}, P)$ be a probability space with $A, B \in \mathcal{A}$. Show that $\mathbb{1}_{A \cap B}(\omega) = \mathbb{1}_A(\omega) \cdot \mathbb{1}_B(\omega)$ for all $\omega \in \Omega$.

Problem 3

Use integration-by-parts to evaluate these integrals.

$\displaystyle \int t^2 e^{-t} \, \textrm{d}t$.
$\displaystyle \int (\ln y)^2 \, \textrm{d}y$.

Problem 4

A computer memory chip fails between times $t_1$ and $t_2$ with probability \[P[ \textrm{fails between $t_1$ and $t_2$} ] = \int_{t_1}^{t_2} \lambda e^{-\lambda x} \textrm{d}x\] where $t_1$ and $t_2$ are times measured in units of hours after start-up and $\lambda$ is a constant with units of $\textrm{hours}^{-1}$.

What is the probability that the chip does not fail in the first $T$ hours?
What is the probability that the chip does fail in the first $T$ hours?
What is the probability that the chip will fail between time $S$ and $S + T$ given that the chip has not failed in the first $S$ hours?

Problem 5

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function \[f(x) = \begin{cases} 5 (1 - x)^4 & \textrm{if } 0 < x < 1 \\ 0 & \textrm{else}, \end{cases}\] what must the capacity of the tank be so that the probability of the supply’s being exhausted in a given week is 0.01?

Problem 6

Consider the function \[f(x) = \begin{cases} k (2x - x^3) & \textrm{if } 0 < x < \frac{5}{2} \\ 0 & \textrm{else}, \end{cases}\] Could $f$ be a probability density function? If so, determine $k$. Repeat for \[f(x) = \begin{cases} k (2x - x^2) & \textrm{if } 0 < x < \frac{5}{2} \\ 0 & \textrm{else}, \end{cases}\]

Problem 7

John is evaluating two radar receivers to measure the position of a target. The measurement errors for Receiver A are thin-tailed and normally distributed around the true target position with $\sigma^2 = 4$. Receiver B uses a cheaper analog front-end that is susceptible to impulsive interference, and its measurement errors are $\text{Cauchy}(0, 1)$. John purchases Receiver B to save money. Mary agrees that both receivers have similar accuracy to within $\pm 2$ units but warns that Receiver B will produce errors beyond $\pm 5$ units at least 10 times as often as Receiver A. Is Mary right?

Problem 8

Suppose that $B$ and $C$ are standard uniform random variables (i.e. $B$ and $C$ $\sim U[0,1]$). Find the probability that $g(x) = x^2 + B \cdot x + C$ has two distinct real roots. If $B = 0.5$ what is the probability that $g(x)$ has single (duplicate) real root? [hint: consider the sign of the discriminant $b^2 - 4ac$ in the quadratic formula].

Problem 9

You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30.

What is the probability that you will have to wait longer than 10 minutes?
If, at 10:15, the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

Problem 10

The random variable $X$ is uniformly distributed in the interval $[0, a]$. Suppose $a$ is unknown so we estimate $a$ by the maximum value observed in $n$ independent repetitions of the experiment; that is $Y = \max(X_1, \ldots, X_n)$. Find $P[Y \le y]$.

Problem 11

Find and plot the pdf of the Weibull random variable: \[F(x) = \begin{cases} 1 - e^{-(x / \lambda)^\beta} & \textrm{if } x \ge 0 \\ 0 & \textrm{else}, \end{cases}\] for $\beta > 0$ and $\lambda > 0$. Choose representative values for $\beta$ and $\lambda$.

Problem 12

The lifetime $X$ of a light bulb is a random variable with \[P[X > t] = \frac{1}{1+t}, \qquad \text{for } t \ge 0.\] Suppose three new light bulbs are installed at time $t = 0$. At time $t = 1$ all three light bulbs are still working. Find the probability that at least one light bulb is still working at time $t = 9$.

Problem 13

Let $X_1, \ldots, X_n$ be independent with common cumulative distribution function $F(x)$. Define $X_{\textrm{max}} = \max(X_1, \ldots, X_N)$ and $X_{\textrm{min}} = \min(X_1, \ldots, X_N)$. Express the cumulative distributions of $X_{\textrm{max}}$ and $X_{\textrm{min}}$ in term of $F(x)$.

Problem 14

An engineer at a radio observatory monitors cosmic background radiation using a five-dish antenna array. The total received noise power from $r$ functioning dishes follows a $\chi^2(r)$ distribution. After a lightning strike, she suspects that two dishes are offline. She records a total noise power of $T = 1$. Assuming it was equally likely that three or five dishes survived the strike, does the data support her suspicion?