Homework #2

EE 364: Spring 2026

Assignment Details

Assigned: 20 January Due: Tuesday, 27 January at 16:00

BrightSpace Assignment: Homework 2

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 #2, Prove the Theorem of Total Probability and Bayes Theorem for evidence $E$ and a partition of hypotheses $H_1, \ldots, H_n$.

Problem 1

Use truth tables to prove whether these propositional assertions are valid or invalid:

$(P \rightarrow \ \sim Q) \Leftrightarrow [(P \vee Q) \wedge \sim (P \vee Q)]$.
$\sim P \Rightarrow (P \Rightarrow Q)$.
$[P \wedge (Q \rightarrow R)] \Leftrightarrow [(\sim P \vee Q) \rightarrow (P \wedge R)]$.
$[(P \rightarrow Q) \wedge (P \rightarrow \ \sim Q)] \Leftrightarrow \ \sim P$.

Problem 2

The sample space describes the set of all possible outcomes of a random experiment. A sigma algebra is a collection of subsets of the sample space, (i.e. a “set of sets”) that is CUT:

Suppose $\Omega$ is a sample space. Then $\mathcal{A}$ is a sigma algebra if $A \in \Omega$ for all $A \in \mathcal{A}$ and

C: if $A \in \mathcal{A}$ then $A^c \in \mathcal{A}$.
U: $A_1, A_2, \ldots \in \mathcal{A}$ implies $\bigcup_k A_k \in \mathcal{A}$.
T: $\Omega \in \mathcal{A}$.

Determine if each of these set collections is a sigma algebra.
1. $\{\emptyset, \Omega\}$.
2. $\{\emptyset, A, A^c, \Omega\}$ where $A \subset \Omega$.
Suppose $\Omega = \{a, b, c, d\}$. Is the set collection $\mathcal{A} = \{\emptyset, \{a\}, \{b\}, \{a, b\}, \{a,c,d\}, \{b,c,d\}, \Omega\}$ a sigma algebra? If not, then could you minimally augment $\mathcal{A}$ so that it is a sigma algebra.

Problem 3

Use mathematical induction to prove these theorems for all positive integers $n \ge 1$:

$11$ divides $23^n - 1$.
$\sqrt[n]{n} < 2 - \frac{1}{n}$ for all $n \ge 2$.

Problem 4

Use the $\epsilon$-definition (i.e. garden hose) to evaluate the limit of these sequences. Given $\epsilon = 10^{-6}$ what is the smallest index $n_0$ such that $|a_n - L| < \epsilon$ for all $n \ge n_0$?

$a_n = e^{-1/\sqrt{n}}$.
$a_n = \frac{4 \sqrt{n}}{3 + \sqrt{n}}$.
$a_n = \ln(n + 1) - \ln(n)$.

Problem 5

Use integration-by-parts to evaluate these integrals.

$\displaystyle \int t^2 e^{-t} dt$.
$\displaystyle \int (\ln y)^2 dy$.

Problem 6

Events $A$ and $B$ are independent with $P(A) > 0$ and $P(B) > 0$. Which one of the following statements must be true.

$A - B \ne \emptyset$.
$A \cap B = \emptyset$.
$A \cap B \ne \emptyset$.
$A \cup B = X$.

Problem 7

Consider the six permutations of the letters $a$, $b$, and $c$ as well as the three triples: $aaa$, $bbb$, and $ccc$. Suppose that these nine triples are points in a sample space and assign probability $1/9$ to each outcome. Let $A_k$ denote the event that the letter $a$ is in the $k$th position and let $B_k$ denote the event that the letter $b$ is in the $k$th position.

Are the events $A_1$ and $A_2$ independent?
Are the events $A_1$, $A_2$, and $A_3$ independent?
Are the events $A_1$ and $B_1$ independent?
Are the events $A_1$ and $B_2$ independent?

Problem 8

Suppose $P(E \cap F) = 0.225$, $P(F) = 0.5$, and $P(E|F^C) = 0.1$. Compute $P(E \cup F)$.

Problem 9

The events $A_1$, $A_2$, and $A_3$ partition sample space $X$. $P(A_1) = 0.3$, $P(A_2) = 0.2$, and $P(A_3) = 0.5$. Suppose $P(E|A_1) = 0.3$, $P(E|A_2) = 0.5$, and $P(E|A_3) = 0.8$. What is the probability $P(A_2|E)$?

Problem 10

Prove or disprove: If $P(A) < P(B)$ and $P(C|A) < P(C|B)$ then $P(A|C) < P(B|C)$.

Problem 11

John picks up a ripe red apple on the ground and wonders whether he should eat it. The apple has rolled down the hillside from where a clump of three apple trees grow. The three apple trees are genetic clones. But the first tree is twice as old as either of the other two apple trees. The first tree also produces twice as many apples as either of the other two trees produce. Only 10% of the apples on the first (older) tree have worms in them because John sprayed the tree with insecticide. He carefully spared the second tree and so only 5% of its apples have worms. John did not spray the third tree and so half of its apples have worms. What is the probability that the apple that John holds has worms?

Problem 12

A meta-analysis selected 100 studies that measured the relation between the consumption of sugary beverages and obesity. 65 of the studies were sponsored by the food industry and 35 studies had no corporate funding. In the sponsored studies: 15% found an unfavorable result, 22% found a neutral result, and 63% found a favorable result. In the non-sponsored studies: 35% found an unfavorable result, 16% found a neutral result, and 49% found a favorable result. What is the probability that a randomly selected study found a favorable result?

Problem 13

Consider a binary symmetric channel (BSC) with $P[\textrm{error}] = 0.1$. Suppose that the symbol “0” is sent 4 times as often as symbol “1”. If you receive $Y = 1$ is it more likely than not that the sent value was “1”?

Problem 14

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 $p_1$. 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 15

Machines A, B, and C manufacture bolts in a factory. Machine A produces 25 percent of the total bolts, B produces 35 percent, and C produces 40 percent. Machine A outputs 5 percent defective bolts, B outputs 4 percent defective bolts, and C outputs 2 percent defective bolts. A bolt is drawn at random from the production line. It is defective. What are the probabilities that the defective bolt was manufactured by machines A, B, and C?

--- title: "Homework #2" 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:** 20 January **Due:** Tuesday, 27 January at 16:00 **BrightSpace Assignment:** [Homework 2](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 #2**, Prove the Theorem of Total Probability and Bayes Theorem for evidence $E$ and a partition of hypotheses $H_1, \ldots, H_n$. ## Problem 1 Use truth tables to prove whether these propositional assertions are valid or invalid: a. $(P \rightarrow \ \sim Q) \Leftrightarrow [(P \vee Q) \wedge \sim (P \vee Q)]$. b. $\sim P \Rightarrow (P \Rightarrow Q)$. c. $[P \wedge (Q \rightarrow R)] \Leftrightarrow [(\sim P \vee Q) \rightarrow (P \wedge R)]$. d. $[(P \rightarrow Q) \wedge (P \rightarrow \ \sim Q)] \Leftrightarrow \ \sim P$. ## Problem 2 The sample space describes the set of all possible outcomes of a random experiment. A sigma algebra is a collection of subsets of the sample space, (i.e. a "set of sets") that is *CUT*: Suppose $\Omega$ is a sample space. Then $\mathcal{A}$ is a sigma algebra if $A \in \Omega$ for all $A \in \mathcal{A}$ and - **C**: if $A \in \mathcal{A}$ then $A^c \in \mathcal{A}$. - **U**: $A_1, A_2, \ldots \in \mathcal{A}$ implies $\bigcup_k A_k \in \mathcal{A}$. - **T**: $\Omega \in \mathcal{A}$. a. Determine if each of these set collections is a sigma algebra. i. $\{\emptyset, \Omega\}$. ii. $\{\emptyset, A, A^c, \Omega\}$ where $A \subset \Omega$. b. Suppose $\Omega = \{a, b, c, d\}$. Is the set collection $\mathcal{A} = \{\emptyset, \{a\}, \{b\}, \{a, b\}, \{a,c,d\}, \{b,c,d\}, \Omega\}$ a sigma algebra? If not, then could you minimally augment $\mathcal{A}$ so that it is a sigma algebra. ## Problem 3 Use mathematical induction to prove these theorems for all positive integers $n \ge 1$: a. $11$ divides $23^n - 1$. b. $\sqrt[n]{n} < 2 - \frac{1}{n}$ for all $n \ge 2$. ## Problem 4 Use the $\epsilon$-definition (i.e. *garden hose*) to evaluate the limit of these sequences. Given $\epsilon = 10^{-6}$ what is the smallest index $n_0$ such that $|a_n - L| < \epsilon$ for all $n \ge n_0$? a. $a_n = e^{-1/\sqrt{n}}$. b. $a_n = \frac{4 \sqrt{n}}{3 + \sqrt{n}}$. c. $a_n = \ln(n + 1) - \ln(n)$. ## Problem 5 Use integration-by-parts to evaluate these integrals. a. $\displaystyle \int t^2 e^{-t} dt$. b. $\displaystyle \int (\ln y)^2 dy$. ## Problem 6 Events $A$ and $B$ are independent with $P(A) > 0$ and $P(B) > 0$. Which **one** of the following statements **must** be true. a. $A - B \ne \emptyset$. b. $A \cap B = \emptyset$. c. $A \cap B \ne \emptyset$. d. $A \cup B = X$. ## Problem 7 Consider the six permutations of the letters $a$, $b$, and $c$ as well as the three triples: $aaa$, $bbb$, and $ccc$. Suppose that these nine triples are points in a sample space and assign probability $1/9$ to each outcome. Let $A_k$ denote the event that the letter $a$ is in the $k$th position and let $B_k$ denote the event that the letter $b$ is in the $k$th position. a. Are the events $A_1$ and $A_2$ independent? b. Are the events $A_1$, $A_2$, and $A_3$ independent? c. Are the events $A_1$ and $B_1$ independent? d. Are the events $A_1$ and $B_2$ independent? ## Problem 8 Suppose $P(E \cap F) = 0.225$, $P(F) = 0.5$, and $P(E|F^C) = 0.1$. Compute $P(E \cup F)$. ## Problem 9 The events $A_1$, $A_2$, and $A_3$ partition sample space $X$. $P(A_1) = 0.3$, $P(A_2) = 0.2$, and $P(A_3) = 0.5$. Suppose $P(E|A_1) = 0.3$, $P(E|A_2) = 0.5$, and $P(E|A_3) = 0.8$. What is the probability $P(A_2|E)$? ## Problem 10 Prove or disprove: If $P(A) < P(B)$ and $P(C|A) < P(C|B)$ then $P(A|C) < P(B|C)$. ## Problem 11 John picks up a ripe red apple on the ground and wonders whether he should eat it. The apple has rolled down the hillside from where a clump of three apple trees grow. The three apple trees are genetic clones. But the first tree is twice as old as either of the other two apple trees. The first tree also produces twice as many apples as either of the other two trees produce. Only 10% of the apples on the first (older) tree have worms in them because John sprayed the tree with insecticide. He carefully spared the second tree and so only 5% of its apples have worms. John did not spray the third tree and so half of its apples have worms. What is the probability that the apple that John holds has worms? ## Problem 12 A meta-analysis selected 100 studies that measured the relation between the consumption of sugary beverages and obesity. 65 of the studies were sponsored by the food industry and 35 studies had no corporate funding. In the sponsored studies: 15% found an unfavorable result, 22% found a neutral result, and 63% found a favorable result. In the non-sponsored studies: 35% found an unfavorable result, 16% found a neutral result, and 49% found a favorable result. What is the probability that a randomly selected study found a favorable result? ## Problem 13 Consider a binary symmetric channel (BSC) with $P[\textrm{error}] = 0.1$. Suppose that the symbol "0" is sent 4 times as often as symbol "1". If you receive $Y = 1$ is it more likely than not that the sent value was "1"? ## Problem 14 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 $p_1$. 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 15 Machines A, B, and C manufacture bolts in a factory. Machine A produces 25 percent of the total bolts, B produces 35 percent, and C produces 40 percent. Machine A outputs 5 percent defective bolts, B outputs 4 percent defective bolts, and C outputs 2 percent defective bolts. A bolt is drawn at random from the production line. It is defective. What are the probabilities that the defective bolt was manufactured by machines A, B, and C?