Homework #5

EE 364: Spring 2026

ImportantAssignment Details

Assigned: 19 February
Due: Thursday, 25 February at 16:00

BrightSpace Assignment: Homework 5

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

Problem 1

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

  1. \(\sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}\)

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

Problem 2

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 \(n \ge n_0\)?

  1. \(a_n = \frac{4 \sqrt{n}}{3 + \sqrt{n}}\).

  2. \(a_n = \ln(n + 1) - \ln(n)\).

Problem 3

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?

Problem 4

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 a slice that is \(\frac{1}{12}\) of the cake.

  1. What is the probability that you get lucky and get no green bits in your slice?

  2. What is the probability that you get really lucky and get no green bits and two or fewer red bits in your slice?

  3. 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 5

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

  1. Find the probability that there are no requests in a 100-ms period.

  2. Find the probability that there are between 5 and 10 requests in a 100-ms period.

Problem 6

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 7

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.

  1. 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].

  2. 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 8

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 9

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

  1. Use a \(\Phi\)-table to calculate \(P[X > 4]\), \(P[X > 7]\), \(P[6.72 < X < 10.16]\), \(P[2 < X < 7]\), and \(P[6 \le X \le 8]\).

  2. If \(P[13 < X \le c] = 0.0123\), find \(c\).

Problem 10

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

  1. What percent of young adults score below 75 on this IQ Test?

  2. What is the probability that a young adult will score above 180 on this IQ Test?

  3. What is the probability that a young adult will score between 85 and 160 on this IQ Test?

  4. 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 11

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 12

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 13

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 14

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 15

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].