Homework #5

EE 364: Spring 2026

Assignment Details

Assigned: 10 February Due: Tuesday, 17 February at 16:00

BrightSpace Assignment: Homework 5

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 #5, Prove that a beta prior $h(\theta) = \textrm{Beta}(\alpha, \beta)$ is conjugate to a binomial likelihood $g(x | \theta) = b(n, \theta)$: $f(\theta | x) = \textrm{Beta}(\alpha + x, \beta + n - x)$.

Problem 1

Steven throws a dart at a dartboard with radius 9 inches. The dart lands randomly on the dartboard at a point with distance $R$ from the center of the dartboard.

Find the probability that the distance from the center $R$ is less than $r$ for $r \ge 0$. [hint: compare the area of the event $R \le r$ to the total area of the dartboard].
Suppose that the bullseye is a circular region with radius 1 inch at the center of the dartboard. Find the probability that the dart lands in the bullseye.

Problem 2

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 3

A system consisting of one original unit plus a spare can function for a random amount of time $X$. If the density of $X$ is given (in units of months) by \[f(x) = \begin{cases} k x e^{-x/2} & \textrm{if } x > 0 \\ 0 & \textrm{else}, \end{cases}\] what is the probability that the system functions for at least 5 months?

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

A firehouse is to be built at some point along a road with length $L$. Fires occur uniformly over the road. If the firehouse is built at a distance a from the left endpoint of the road what is the expected distance the firetruck will have to travel?

Problem 7

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 8

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 9

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 10

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 11

Let X be a Gaussian random variable with $\mu = 5$ and $\sigma^2 = 16$.

Use a $\Phi$-table to calculate $P[X > 4]$, $P[X > 7]$, $P[6.72 < X < 10.16]$, $P[2 < X < 7]$, and $P[6 \le X \le 8]$.
If $P[13 < X \le c] = 0.0123$, find $c$.

Problem 12

Scores on a standard IQ Test for people aged 20 to 34 are normally distributed with mean 110 and standard deviation 25.

What percent of young adults score below 75 on this IQ Test?
What is the probability that a young adult will score above 180 on this IQ Test?
What is the probability that a young adult will score between 85 and 160 on this IQ Test?
A university only wants to accept individuals in the top 15% of this intelligence scale. How high must a person score to be considered for admission?

Problem 13

Let $X$ be a normal random variable with mean 12 and variance 4. Find the value of $c$ such that $P[X > c] = 0.10$.

Problem 14

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will give a reading that is normally distributed with $\mu = 4$ and $\sigma^2 = 4$, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with $\mu = 6$ and $\sigma^2 = 9$. A point is randomly chosen on the image and has a reading of 5. If the fraction of the image that is black is $\alpha$, for what value of $\alpha$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?

Problem 15

Two chips are being considered for use in a certain system. The lifetime of chip 1 is modeled by a Gaussian random variable with mean 20,000 hours and standard deviation 5000 hours (the probability of negative lifetime is negligible). The lifetime of chip 2 is also a Gaussian random variable but with mean 22,000 hours and standard deviation 1000 hours. Which chip is preferred if the target lifetime of the system is 20,000 hours? 24,000 hours?

Problem 16

The mode of a random variable $X$ is $x_M = \displaystyle{\arg\max_x} f_X(x)$. The full width at half maximum for unimodal random variable $X$ is defined as \[\textrm{FWHM}(X) = x_2 - x_1,\] where $x_1 < x_M < x_2$ such that $f_X(x_1) = f_X(x_2) = \frac12 f_X(x_M)$. Compute the FWHM for $X \sim N(\mu, \sigma^2)$.

Problem 17

You send binary random voltage $X$ (0V or 5V) through an asymmetric noisy channel. A detector receives the corrupted signal $Y = X + N$ and applies a simple threshold $0 < T < 5$ to estimate the binary input $X$. Suppose that the noise is zero-mean Gaussian whose variance depends on the input $X$: $N_{|X=0} \sim N(0,4)$ and $N_{|X=1} \sim N(0,9)$. What value of $T$ ensures that the probability of Type 1 error is equal to the probability of Type 2 error? [hint: do not use the normal-CDF table].

Problem 18

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 19

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 20

Suppose Bayesian prior $p_k(\theta)$ and posterior $p_k(\theta|x)$ are conjugate with likelihood $f(x|\theta)$ for $k = \{1, \ldots, n\}$. Show that the mixture prior $p(\theta) = \sum_{k=1}^{n} w_k p_k(\theta)$ is conjugate and find the conjugate posterior $p(\theta|x)$ as a function of $p_k(\theta|x)$ — mixing weights $0 < w_k < 1$ and $\sum_{k=1}^{n} w_k = 1$. Use this fact to find the posterior of the success probability $p$ after performing twelve coin-flips and observing 9 successes under prior $p \sim 0.25 \textrm{Beta}(1,1) + 0.75 \textrm{Beta}(5,5)$.

--- title: "Homework #5" subtitle: "EE 364: Spring 2026" sidebar: main format: html: toc: false number-sections: false code-fold: show code-tools: true theme: cosmo --- ::: {.callout-important} ## Assignment Details **Assigned:** 10 February **Due:** Tuesday, 17 February at 16:00 **BrightSpace Assignment:** [Homework 5](https://brightspace.usc.edu) ::: ::: {.callout-warning} ## 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 #5**, Prove that a beta prior $h(\theta) = \textrm{Beta}(\alpha, \beta)$ is conjugate to a binomial likelihood $g(x | \theta) = b(n, \theta)$: $f(\theta | x) = \textrm{Beta}(\alpha + x, \beta + n - x)$. ## Problem 1 Steven throws a dart at a dartboard with radius 9 inches. The dart lands randomly on the dartboard at a point with distance $R$ from the center of the dartboard. a. Find the probability that the distance from the center $R$ is less than $r$ for $r \ge 0$. [hint: compare the area of the event $R \le r$ to the total area of the dartboard]. b. Suppose that the bullseye is a circular region with radius 1 inch at the center of the dartboard. Find the probability that the dart lands in the bullseye. ## Problem 2 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 3 A system consisting of one original unit plus a spare can function for a random amount of time $X$. If the density of $X$ is given (in units of months) by $$f(x) = \begin{cases} k x e^{-x/2} & \textrm{if } x > 0 \\ 0 & \textrm{else}, \end{cases}$$ what is the probability that the system functions for at least 5 months? ## 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 $\textrm{hours}^{-1}$. a. What is the probability that the chip does not fail in the first $T$ hours? b. What is the probability that the chip does fail in the first $T$ hours? c. 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 A firehouse is to be built at some point along a road with length $L$. Fires occur uniformly over the road. If the firehouse is built at a distance a from the left endpoint of the road what is the expected distance the firetruck will have to travel? ## Problem 7 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 8 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. a. What is the probability that you will have to wait longer than 10 minutes? b. 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 9 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 10 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 11 Let X be a Gaussian random variable with $\mu = 5$ and $\sigma^2 = 16$. a. Use a $\Phi$-table to calculate $P[X > 4]$, $P[X > 7]$, $P[6.72 < X < 10.16]$, $P[2 < X < 7]$, and $P[6 \le X \le 8]$. b. If $P[13 < X \le c] = 0.0123$, find $c$. ## Problem 12 Scores on a standard IQ Test for people aged 20 to 34 are normally distributed with mean 110 and standard deviation 25. a. What percent of young adults score below 75 on this IQ Test? b. What is the probability that a young adult will score above 180 on this IQ Test? c. What is the probability that a young adult will score between 85 and 160 on this IQ Test? d. A university only wants to accept individuals in the top 15% of this intelligence scale. How high must a person score to be considered for admission? ## Problem 13 Let $X$ be a normal random variable with mean 12 and variance 4. Find the value of $c$ such that $P[X > c] = 0.10$. ## Problem 14 An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will give a reading that is normally distributed with $\mu = 4$ and $\sigma^2 = 4$, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with $\mu = 6$ and $\sigma^2 = 9$. A point is randomly chosen on the image and has a reading of 5. If the fraction of the image that is black is $\alpha$, for what value of $\alpha$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region? ## Problem 15 Two chips are being considered for use in a certain system. The lifetime of chip 1 is modeled by a Gaussian random variable with mean 20,000 hours and standard deviation 5000 hours (the probability of negative lifetime is negligible). The lifetime of chip 2 is also a Gaussian random variable but with mean 22,000 hours and standard deviation 1000 hours. Which chip is preferred if the target lifetime of the system is 20,000 hours? 24,000 hours? ## Problem 16 The mode of a random variable $X$ is $x_M = \displaystyle{\arg\max_x} f_X(x)$. The *full width at half maximum* for unimodal random variable $X$ is defined as $$\textrm{FWHM}(X) = x_2 - x_1,$$ where $x_1 < x_M < x_2$ such that $f_X(x_1) = f_X(x_2) = \frac12 f_X(x_M)$. Compute the FWHM for $X \sim N(\mu, \sigma^2)$. ## Problem 17 You send binary random voltage $X$ (0V or 5V) through an asymmetric noisy channel. A detector receives the corrupted signal $Y = X + N$ and applies a simple threshold $0 < T < 5$ to estimate the binary input $X$. Suppose that the noise is zero-mean Gaussian whose variance depends on the input $X$: $N_{|X=0} \sim N(0,4)$ and $N_{|X=1} \sim N(0,9)$. What value of $T$ ensures that the probability of Type 1 error is equal to the probability of Type 2 error? [hint: do not use the normal-CDF table]. ## Problem 18 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 19 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 20 Suppose Bayesian prior $p_k(\theta)$ and posterior $p_k(\theta|x)$ are conjugate with likelihood $f(x|\theta)$ for $k = \{1, \ldots, n\}$. Show that the mixture prior $p(\theta) = \sum_{k=1}^{n} w_k p_k(\theta)$ is conjugate and find the conjugate posterior $p(\theta|x)$ as a function of $p_k(\theta|x)$ --- mixing weights $0 < w_k < 1$ and $\sum_{k=1}^{n} w_k = 1$. Use this fact to find the posterior of the success probability $p$ after performing twelve coin-flips and observing 9 successes under prior $p \sim 0.25 \textrm{Beta}(1,1) + 0.75 \textrm{Beta}(5,5)$.