Homework #6

EE 364: Spring 2026

Assignment Details

Assigned: 17 February Due: Tuesday, 24 February 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, Derive all the BEG-CUP pdf moments: binomial, geometric, hypergeometric, negative binomial, Poisson, gamma, exponential, chi-square, beta, uniform, Gaussian.

Problem 1

Let $A$ be an event. Define the indicator function \[I_A(x) = \begin{cases} 1 & \textrm{if } x \in A \\ 0 & \textrm{if } x \notin A. \end{cases}\]

Evaluate $I_A(x) + I_{A^C}(x)$.
Compute $E[I_A]$.
Prove: $E[X] \ge E[X \cdot I_{\{X \ge c\}}]$ for any constant $c$ and any positive random variable $X$ with $E[X] < \infty$.

Problem 2

Let $Y = A \, \cos(\omega t) + c$ where $\omega$ and $c$ are constants with $E[A] = \mu$ and $V[A] = \sigma^2$. Find the mean and variance of $Y$.

Problem 3

Find the mean and variance of $X$ if

\[f_X(x) = \begin{cases} c (1-x^2) & -1 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}\]
\[f_X(x) = \begin{cases} c x (1-x^2) & 0 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}\]

Problem 4

Suppose that $B$ and $C$ are standard uniform random variables, i.e. $B \sim U[0,1]$ and $C \sim U[0,1]$. Calculate $E[B^n]$ and $V[B^n]$.

Problem 5

A dart is equally likely to land at any point inside a circular target of radius 2. Let $R$ be the distance of the landing point from the origin. Find the mean and variance of $R$.

Problem 6

Random variable $X$ equals 2 with probability $0.4$ and is uniformly distributed $[0, 1]$ otherwise. What is $E[X] + V[X]$?

Problem 7

Suppose that $A$ and $B$ each randomly and independently choose 3 of 10 objects. Find the expected number of objects,

chosen by both $A$ and $B$.
not chosen by either $A$ or $B$.
chosen by exactly one of $A$ and $B$.

Problem 8

Let $Z \sim N(0, 1)$. Define $Y = Z + n$ for some constant $n$. Compute $E[Y^6]$.

Problem 9

Random variable $X \sim N(0, 4)$. What is $E[|X|]$?

Problem 10

Suppose $Y$ is a non-negative random variable.

Show: $\displaystyle E[Y] = \int_{0}^{\infty} P(Y > t) dt$.
Show: $\displaystyle E[Y^n] = \int_{0}^{\infty} n \, x^{n-1} P(Y > x) dx$.

Problem 11

Random variable $X_1 \sim N(0, 1)$. Define $X_i \sim N(X_{i-1}, 1)$. What is the distribution of $X_n$?

Problem 12

Let $X$ be a Pareto random variable, $f_X(x) = \frac{2}{x^3}$ for $x \ge 1$, and $f(x)=0$ otherwise.

Compute $E[X]$.
Compute $E[X^2]$.

Problem 13

Let $X$ be a Poisson random variable with parameter $\lambda$. Show directly that $E[X^n] = \lambda E[( X + 1 )^{n - 1}]$. Use the result to compute $E[X^3]$.

Problem 14

Suppose that $X$ is a random variable with pdf \[f_X(x) = \begin{cases} a + b x^2 & 0 \le x \le 1 \\ 0 & \textrm{ else}. \end{cases}\] Find $a$ and $b$ if $E[X] = \frac{3}{5}$.

Problem 15

Consider the following limiter, $g(x)$:

Find expressions for the mean and variance of $Y = g(X)$ for an arbitrary continuous random variable $X$ with symmetric pdf, $f_X(x)$.
Suppose $a = 1$. Evaluate the mean and variance if $X$ is a Laplacian random variable with $\lambda = 1$, that is $f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)$.
Suppose $a = \frac12$. Evaluate the mean and variance if $X = U^3$ where $U \sim U[-1, 1]$.

Problem 16

Consider the following limiter with center-level clipping, $h(x)$:

Find expressions for the mean and variance of $Y = h(X)$ for an arbitrary continuous random variable $X$ with symmetric pdf, $f_X(-x) = f_X(x)$.
Suppose $a = 1$ and $b = 2$. Evaluate the mean and variance if $X$ is a Laplacian random variable with $\lambda = 1$, that is $f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)$.
Suppose $a = \frac12$ and $b = \frac34$. Evaluate the mean and variance if $X = b \cos(2 \pi U)$ where $U \sim U[-1, 1]$.

--- title: "Homework #6" 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:** 17 February **Due:** Tuesday, 24 February at 16:00 **BrightSpace Assignment:** [Homework 6](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 #6**, Derive all the BEG-CUP pdf moments: binomial, geometric, hypergeometric, negative binomial, Poisson, gamma, exponential, chi-square, beta, uniform, Gaussian. ## Problem 1 Let $A$ be an event. Define the indicator function $$I_A(x) = \begin{cases} 1 & \textrm{if } x \in A \\ 0 & \textrm{if } x \notin A. \end{cases}$$ a. Evaluate $I_A(x) + I_{A^C}(x)$. b. Compute $E[I_A]$. c. Prove: $E[X] \ge E[X \cdot I_{\{X \ge c\}}]$ for any constant $c$ and any positive random variable $X$ with $E[X] < \infty$. ## Problem 2 Let $Y = A \, \cos(\omega t) + c$ where $\omega$ and $c$ are constants with $E[A] = \mu$ and $V[A] = \sigma^2$. Find the mean and variance of $Y$. ## Problem 3 Find the mean and variance of $X$ if a. $$f_X(x) = \begin{cases} c (1-x^2) & -1 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}$$ b. $$f_X(x) = \begin{cases} c x (1-x^2) & 0 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}$$ ## Problem 4 Suppose that $B$ and $C$ are standard uniform random variables, i.e. $B \sim U[0,1]$ and $C \sim U[0,1]$. Calculate $E[B^n]$ and $V[B^n]$. ## Problem 5 A dart is equally likely to land at any point inside a circular target of radius 2. Let $R$ be the distance of the landing point from the origin. Find the mean and variance of $R$. ## Problem 6 Random variable $X$ equals 2 with probability $0.4$ and is uniformly distributed $[0, 1]$ otherwise. What is $E[X] + V[X]$? ## Problem 7 Suppose that $A$ and $B$ each randomly and independently choose 3 of 10 objects. Find the expected number of objects, a. chosen by both $A$ and $B$. b. not chosen by either $A$ or $B$. c. chosen by exactly one of $A$ and $B$. ## Problem 8 Let $Z \sim N(0, 1)$. Define $Y = Z + n$ for some constant $n$. Compute $E[Y^6]$. ## Problem 9 Random variable $X \sim N(0, 4)$. What is $E[|X|]$? ## Problem 10 Suppose $Y$ is a non-negative random variable. a. Show: $\displaystyle E[Y] = \int_{0}^{\infty} P(Y > t) dt$. b. Show: $\displaystyle E[Y^n] = \int_{0}^{\infty} n \, x^{n-1} P(Y > x) dx$. ## Problem 11 Random variable $X_1 \sim N(0, 1)$. Define $X_i \sim N(X_{i-1}, 1)$. What is the distribution of $X_n$? ## Problem 12 Let $X$ be a Pareto random variable, $f_X(x) = \frac{2}{x^3}$ for $x \ge 1$, and $f(x)=0$ otherwise. a. Compute $E[X]$. b. Compute $E[X^2]$. ## Problem 13 Let $X$ be a Poisson random variable with parameter $\lambda$. Show directly that $E[X^n] = \lambda E[( X + 1 )^{n - 1}]$. Use the result to compute $E[X^3]$. ## Problem 14 Suppose that $X$ is a random variable with pdf $$f_X(x) = \begin{cases} a + b x^2 & 0 \le x \le 1 \\ 0 & \textrm{ else}. \end{cases}$$ Find $a$ and $b$ if $E[X] = \frac{3}{5}$. ## Problem 15 Consider the following limiter, $g(x)$: ```{python} #| echo: false #| fig-align: center import matplotlib.pyplot as plt import numpy as np fig, ax = plt.subplots(figsize=(5, 4)) # Define the limiter function x = np.linspace(-3.5, 3.5, 1000) a = 1.5 y = np.clip(x, -a, a) ax.plot(x, y, 'b-', linewidth=2) ax.axhline(y=0, color='k', linewidth=0.5) ax.axvline(x=0, color='k', linewidth=0.5) # Dashed lines ax.plot([-a, -a], [0, -a], 'k--', linewidth=0.5) ax.plot([0, -a], [-a, -a], 'k--', linewidth=0.5) ax.plot([a, a], [0, a], 'k--', linewidth=0.5) ax.plot([0, a], [a, a], 'k--', linewidth=0.5) ax.set_xlabel('$x$', fontsize=12) ax.set_ylabel('$g(x)$', fontsize=12) ax.set_xticks([-a, a]) ax.set_xticklabels(['$-a$', '$a$']) ax.set_yticks([-a, a]) ax.set_yticklabels(['$-a$', '$a$']) ax.set_xlim(-3.5, 3.5) ax.set_ylim(-2.5, 2.5) ax.set_aspect('equal') plt.tight_layout() plt.show() ``` a. Find expressions for the mean and variance of $Y = g(X)$ for an arbitrary continuous random variable $X$ with **symmetric** pdf, $f_X(x)$. b. Suppose $a = 1$. Evaluate the mean and variance if $X$ is a Laplacian random variable with $\lambda = 1$, that is $f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)$. c. Suppose $a = \frac12$. Evaluate the mean and variance if $X = U^3$ where $U \sim U[-1, 1]$. ## Problem 16 Consider the following limiter with center-level clipping, $h(x)$: ```{python} #| echo: false #| fig-align: center import matplotlib.pyplot as plt import numpy as np fig, ax = plt.subplots(figsize=(5, 4)) a = 0.5 b = 1.5 # Define the piecewise function x1 = np.linspace(-3.5, -b, 100) y1 = np.full_like(x1, -b) x2 = np.linspace(-b, -a, 100) y2 = (b / (b - a)) * (x2 + a) x3 = np.linspace(-a, a, 100) y3 = np.zeros_like(x3) x4 = np.linspace(a, b, 100) y4 = (b / (b - a)) * (x4 - a) x5 = np.linspace(b, 3.5, 100) y5 = np.full_like(x5, b) ax.plot(np.concatenate([x1, x2, x3, x4, x5]), np.concatenate([y1, y2, y3, y4, y5]), 'b-', linewidth=2) ax.axhline(y=0, color='k', linewidth=0.5) ax.axvline(x=0, color='k', linewidth=0.5) # Dashed lines ax.plot([-b, -b], [0, -b], 'k--', linewidth=0.5) ax.plot([0, -b], [-b, -b], 'k--', linewidth=0.5) ax.plot([b, b], [0, b], 'k--', linewidth=0.5) ax.plot([0, b], [b, b], 'k--', linewidth=0.5) ax.set_xlabel('$x$', fontsize=12) ax.set_ylabel('$h(x)$', fontsize=12) ax.set_xticks([-b, -a, a, b]) ax.set_xticklabels(['$-b$', '$-a$', '$a$', '$b$']) ax.set_yticks([-b, b]) ax.set_yticklabels(['$-b$', '$b$']) ax.set_xlim(-3.5, 3.5) ax.set_ylim(-2.5, 2.5) ax.set_aspect('equal') plt.tight_layout() plt.show() ``` a. Find expressions for the mean and variance of $Y = h(X)$ for an arbitrary continuous random variable $X$ with **symmetric** pdf, $f_X(-x) = f_X(x)$. b. Suppose $a = 1$ and $b = 2$. Evaluate the mean and variance if $X$ is a Laplacian random variable with $\lambda = 1$, that is $f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)$. c. Suppose $a = \frac12$ and $b = \frac34$. Evaluate the mean and variance if $X = b \cos(2 \pi U)$ where $U \sim U[-1, 1]$.