Homework #4

EE 364: Spring 2026

ImportantAssignment Details

Assigned: 05 February
Due: Thursday, 12 February at 16:00

BrightSpace Assignment: Homework 4

WarningInstructions

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 Trinomial Theorem. Derive the trinomial PMF \(p(x, y)\). Show that the marginal \(p(x)\) and conditional \(p(x \mid y)\) are both binomial.

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 \(f\): \(f^{-1}(S) = \{z \in \mathbb{Z} : f(z) \in S\}\).

  1. Find \(f^{-1}(A)\), \(f^{-1}(B)\), \(f^{-1}(A \cup B)\), and \(f^{-1}(A \cap B)\).

  2. Verify the commutations \(f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)\) and \(f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)\).

  3. 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:

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

  2. \(\sum_{n=1}^{\infty} \frac{n^{\pi}}{\pi^n}\).

  3. \(\sum_{n=1}^{\infty} \left(\frac{n^2 + 1}{2n^2 + 1}\right)^n\).

Problem 3

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

Problem 4

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

  1. Find the probability that \(M\) is odd.

  2. 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 5

Patients are recruited into two arms of a clinical trial: control and treatment. The probability that an adverse outcome occurs for an individual in the control group is \(p_0\). The probability that an adverse outcome occurs for an individual in the treatment group is \(p_1\). Patients are allocated in strict alternating order (control, treatment, control, treatment, …). What is the probability that the first adverse event occurs on the control arm? Assume that outcomes are independent.

Problem 6

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

  1. \(P(\min(X_1, \ldots, X_n) > a)\).

  2. \(P(\max(X_1, \ldots, X_n) \le a)\).

Problem 7

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 8

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 9

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 10

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 11

A quality control inspector samples 12 items from a production line with replacement. Each item is independently classified as “pass” with probability 0.7, “minor defect” with probability 0.2, or “major defect” with probability 0.1. Let \(X\) denote the number of items that pass and \(Y\) denote the number of items with minor defects.

  1. What is the probability that exactly 8 items pass and exactly 3 have minor defects?

  2. What is the probability that exactly 8 items pass (regardless of the defect breakdown)?

  3. Given that exactly 8 items pass, what is the probability that exactly 3 of the remaining items have minor defects?

Problem 12

A six-sided die is rolled 9 times.

  1. What is the probability of obtaining exactly three 1’s, two 2’s, two 3’s, one 4, one 5, and zero 6’s?

  2. What is the probability of obtaining exactly three of one face, two each of two other faces, and one each of the remaining two faces?

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?