Homework #4

EE 364: Spring 2026

Assignment Details

Assigned: 03 February Due: Tuesday, 10 February at 16:00

BrightSpace Assignment: Homework 4

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 #4, Prove the Poisson Law: $b(n, p) \rightarrow P(\lambda)$ if $\lambda = n p$.

Problem 1

Consider the integer function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(n) = n^2 + 4$ for any integer $n$ in the set of integers $\mathbb{Z}$. Define the range subsets $A = \{4, 5, 13, 20\} \subset \mathbb{Z}$ and $B = \{8, 13\} \subset \mathbb{Z}$. Define the pullback or inverse image set $f^{-1}(S)$ as the set of pre-images $z \in \mathbb{Z}$ of $S$ under the mapping $f$: $f^{-1}(S) = \{z \in \mathbb{Z} : f(z) \in S\}$.

Find $f^{-1}(A)$, $f^{-1}(B)$, $f^{-1}(A \cup B)$, and $f^{-1}(A \cap B)$.
Verify the commutations $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$ and $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$.
Is it also true that $f(A \cup B) = f(A) \cup f(B)$ and $f(A \cap B) = f(A) \cap f(B)$?

Problem 2

Use the ratio test to determine whether the following infinite series diverge or converge:

$\sum_{n=1}^{\infty} \frac{10^n}{(n+1) 4^{2n+1}}$.
$\sum_{n=1}^{\infty} \frac{n^{\pi}}{\pi^n}$.
$\sum_{n=1}^{\infty} \left(\frac{n^2 + 1}{2n^2 + 1}\right)^n$.

Problem 3

Find the interval of convergence for these power series (check both endpoints):

$\sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}$
$\sum_{n=1}^{\infty} \frac{\ln n}{n} x^n$
$\sum_{n=1}^{\infty} \frac{(n!)^2}{n^n} (x - 2)^n$

Problem 4

We do not know how the Ancient Greek engineer Archimedes proved his famous result $\frac{265}{153} < \sqrt{3} < \frac{1351}{780}$ described in “Measurement of a Circle”. Use a proof by contradiction to show that $\sqrt{3}$ is an irrational number.

Problem 5

Flip a fair coin. What is the probability that you will need at least five flips until you get two heads?

Problem 6

Let $M$ be a geometric random variable with $S_M = \{1,2,\cdots\}$.

Find the probability that $M$ is odd.
Show that $M$ satisfies the memoryless property: $P(M \ge k + j | M \ge j + 1) = P(M \ge k)$ for all $j, k > 1$.

Problem 7

A Christmas fruitcake has Poisson-distributed independent numbers of sultana raisins, iridescent red cherry bits, and radioactive green cherry bits with respective averages 48, 24, and 12 bits per cake. Suppose you politely accept $\frac{1}{12}$ of a slice of the cake.

What is the probability that you get lucky and get no green bits in your slice?
What is the probability that you get really lucky and get no green bits and two or fewer red bits in your slice?
What is the probability that you get extremely lucky and get no green or red bits and more than five raisins in your slice?

Problem 8

The number of page requests that arrive at a web server is a Poisson random variable with an average of 6000 requests per minute.

Find the probability that there are no requests in a 100-ms period.
Find the probability that there are between 5 and 10 requests in a 100-ms period.

Problem 9

Random variables $X_1, \ldots, X_n$ are independent geometric random variables with success probability $p$. Suppose $a > 0$. Compute the following probabilities.

$P(\min(X_1, \ldots, X_n) > a)$.
$P(\max(X_1, \ldots, X_n) \le a)$.

Problem 10

John claims that a geometric random variable $X$ has a constant hazard rate $h(k)$ where \[h(k) = \frac{p(k)}{1 - F(k-1)}.\] The hazard rate $h(k)$ is the probability that the process will fail on the $k$th trial given that it has not failed on the first $k-1$ trials. Here $p(k) = P(X = k)$ is the probability mass function (PMF) of $X$ and $F(k) = P(X \le k)$ is the cumulative distribution function (CDF) of $X$. Do you agree with John’s claim?

Problem 11

John is choosing between two models to count independent customer arrivals to his website. The first model uses parameter $\lambda = 2$. The second model uses $\lambda = 3$. John believes that the second model is twice as probable as the first, i.e. $P(\lambda = 2) = \frac13$ and $P(\lambda = 3) = \frac23$. The only available data consists of two random (independent and identically distributed) samples: $x_1 = 2$ and $x_2 = 4$. Which of John’s two models is the most probable given these facts?

Problem 12

Mary performs a psychological survey with 20-college age students. She randomly distributes the 20 students into four groups with five students each. Study participants independently withdraw with 20% probability each month. What is the probability that after the first month at least two groups have four or more members?

Problem 13

In a close election between two candidates A and B in a small town the winning margin of candidate A is 1422 to 1405 votes. But 101 votes are illegal and will be thrown out. Assume that the illegal votes are not biased in any particular way (that is, $P[\textrm{vote A}] = P[\textrm{vote B}]$). What is the probability that removing the illegal votes will change the result of the election?

Problem 14

John starts a new Internet company that sells high-end golf balls made in a German factory. The golf balls come in a box of twelve. John has introduced his own trademarked line of expensive balls but finds that only a quarter of customers buy them. He still manages to sell several of the trademarked boxes each day. It is now Sunday night and John looks forward to the next day’s flow of customer orders. What is the probability that the ninth customer will be the fifth customer to order the new trademarked golf balls?

Problem 15

Mary and John compete in a trivia game involving a series of questions. The probability that Mary gives the correct answer on any question is $\alpha$ and the probability that John gives the correct answer on any question is $\beta$. Assume that the outcome of each question is independent of all other questions. A player wins if they answer a question correctly. Compute the probability that John will win if Mary answers the first question. Simplify your answer.

Problem 16

A infinite discrete probability density $\{p_k\}$ obeys $\sum_{k=1}^{\infty} p_k = 1$ where $p_k = P(X = k) \ge 0$ for every positive integer $k$. The density has a probability generating function (PGF) $G_X(s) = \sum_{k=1}^{\infty} p_k s^k$ for some interval of convergence of $s$ values. The geometric probability density measures the probability that it takes k Bernoulli trials until the first success occurs: $p_k = P(X = k) = q^{k-1} p$ where $q = 1 - p$ for success probability $p$. What is the geometric PGF? What is the interval of convergence for the geometric PGF?

--- title: "Homework #4" 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:** 03 February **Due:** Tuesday, 10 February at 16:00 **BrightSpace Assignment:** [Homework 4](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 #4**, Prove the Poisson Law: $b(n, p) \rightarrow P(\lambda)$ if $\lambda = n p$. ## Problem 1 Consider the integer function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(n) = n^2 + 4$ for any integer $n$ in the set of integers $\mathbb{Z}$. Define the range subsets $A = \{4, 5, 13, 20\} \subset \mathbb{Z}$ and $B = \{8, 13\} \subset \mathbb{Z}$. Define the pullback or *inverse image set* $f^{-1}(S)$ as the set of pre-images $z \in \mathbb{Z}$ of $S$ under the mapping $f$: $f^{-1}(S) = \{z \in \mathbb{Z} : f(z) \in S\}$. a. Find $f^{-1}(A)$, $f^{-1}(B)$, $f^{-1}(A \cup B)$, and $f^{-1}(A \cap B)$. b. Verify the commutations $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$ and $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$. c. Is it also true that $f(A \cup B) = f(A) \cup f(B)$ and $f(A \cap B) = f(A) \cap f(B)$? ## Problem 2 Use the ratio test to determine whether the following infinite series diverge or converge: a. $\sum_{n=1}^{\infty} \frac{10^n}{(n+1) 4^{2n+1}}$. b. $\sum_{n=1}^{\infty} \frac{n^{\pi}}{\pi^n}$. c. $\sum_{n=1}^{\infty} \left(\frac{n^2 + 1}{2n^2 + 1}\right)^n$. ## Problem 3 Find the interval of convergence for these power series (check both endpoints): a. $\sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}$ b. $\sum_{n=1}^{\infty} \frac{\ln n}{n} x^n$ c. $\sum_{n=1}^{\infty} \frac{(n!)^2}{n^n} (x - 2)^n$ ## Problem 4 We do not know how the Ancient Greek engineer Archimedes proved his famous result $\frac{265}{153} < \sqrt{3} < \frac{1351}{780}$ described in "Measurement of a Circle". Use a proof by contradiction to show that $\sqrt{3}$ is an irrational number. ## Problem 5 Flip a fair coin. What is the probability that you will need at least five flips until you get two heads? ## Problem 6 Let $M$ be a geometric random variable with $S_M = \{1,2,\cdots\}$. a. Find the probability that $M$ is odd. b. Show that $M$ satisfies the memoryless property: $P(M \ge k + j | M \ge j + 1) = P(M \ge k)$ for all $j, k > 1$. ## Problem 7 A Christmas fruitcake has Poisson-distributed independent numbers of sultana raisins, iridescent red cherry bits, and radioactive green cherry bits with respective averages 48, 24, and 12 bits per cake. Suppose you politely accept $\frac{1}{12}$ of a slice of the cake. a. What is the probability that you get lucky and get no green bits in your slice? b. What is the probability that you get really lucky and get no green bits and two or fewer red bits in your slice? c. What is the probability that you get extremely lucky and get no green or red bits and more than five raisins in your slice? ## Problem 8 The number of page requests that arrive at a web server is a Poisson random variable with an average of 6000 requests per minute. a. Find the probability that there are no requests in a 100-ms period. b. Find the probability that there are between 5 and 10 requests in a 100-ms period. ## Problem 9 Random variables $X_1, \ldots, X_n$ are independent geometric random variables with success probability $p$. Suppose $a > 0$. Compute the following probabilities. a. $P(\min(X_1, \ldots, X_n) > a)$. b. $P(\max(X_1, \ldots, X_n) \le a)$. ## Problem 10 John claims that a geometric random variable $X$ has a constant hazard rate $h(k)$ where $$h(k) = \frac{p(k)}{1 - F(k-1)}.$$ The hazard rate $h(k)$ is the probability that the process will fail on the $k$th trial given that it has not failed on the first $k-1$ trials. Here $p(k) = P(X = k)$ is the *probability mass function* (PMF) of $X$ and $F(k) = P(X \le k)$ is the *cumulative distribution function* (CDF) of $X$. Do you agree with John's claim? ## Problem 11 John is choosing between two models to count independent customer arrivals to his website. The first model uses parameter $\lambda = 2$. The second model uses $\lambda = 3$. John believes that the second model is twice as probable as the first, i.e. $P(\lambda = 2) = \frac13$ and $P(\lambda = 3) = \frac23$. The only available data consists of two random (independent and identically distributed) samples: $x_1 = 2$ and $x_2 = 4$. Which of John's two models is the most probable given these facts? ## Problem 12 Mary performs a psychological survey with 20-college age students. She randomly distributes the 20 students into four groups with five students each. Study participants independently withdraw with 20% probability each month. What is the probability that after the first month at least two groups have four or more members? ## Problem 13 In a close election between two candidates A and B in a small town the winning margin of candidate A is 1422 to 1405 votes. But 101 votes are illegal and will be thrown out. Assume that the illegal votes are not biased in any particular way (that is, $P[\textrm{vote A}] = P[\textrm{vote B}]$). What is the probability that removing the illegal votes will change the result of the election? ## Problem 14 John starts a new Internet company that sells high-end golf balls made in a German factory. The golf balls come in a box of twelve. John has introduced his own trademarked line of expensive balls but finds that only a quarter of customers buy them. He still manages to sell several of the trademarked boxes each day. It is now Sunday night and John looks forward to the next day's flow of customer orders. What is the probability that the ninth customer will be the fifth customer to order the new trademarked golf balls? ## Problem 15 Mary and John compete in a trivia game involving a series of questions. The probability that Mary gives the correct answer on any question is $\alpha$ and the probability that John gives the correct answer on any question is $\beta$. Assume that the outcome of each question is independent of all other questions. A player wins if they answer a question correctly. Compute the probability that John will win if Mary answers the first question. Simplify your answer. ## Problem 16 A infinite discrete probability density $\{p_k\}$ obeys $\sum_{k=1}^{\infty} p_k = 1$ where $p_k = P(X = k) \ge 0$ for every positive integer $k$. The density has a *probability generating function* (PGF) $G_X(s) = \sum_{k=1}^{\infty} p_k s^k$ for some interval of convergence of $s$ values. The geometric probability density measures the probability that it takes k Bernoulli trials until the first success occurs: $p_k = P(X = k) = q^{k-1} p$ where $q = 1 - p$ for success probability $p$. What is the geometric PGF? What is the interval of convergence for the geometric PGF?