Homework #11
EE 364: Spring 2026
Assigned: 28 April
Due: Thursday, 05 May at 16:00
BrightSpace Assignment: Homework 11
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 #11
- Prove the Central Limit Theorem: \(Z_n = \text{STD}(\bar{X}_n) \xrightarrow{d} Z \sim N(0,1)\).
- Derive all MGFs on the BEG-CUP pdf sheet.
Problem 1
Derive the moment generating function of a negative binomial random variable \(X\). Use the result to compute \(E[X]\) and \(V[X]\).
Problem 2
Random variables \(X_1, \ldots, X_n\) are independent with \(X_k \sim \text{Gamma}(\alpha_k, \theta)\) for fixed \(\theta > 0\). Identify the distribution of \(Y_n = \sum_{k=1}^n X_k\). Use the result to identify the distribution of \(\sum_{k=1}^n Z_k\) for i.i.d. \(Z_k \sim \chi^2(1)\).
Problem 3
Random variables \(X\) and \(Y\) are zero-mean Gaussian with \(X \sim N(0, \sigma_X^2)\) and \(Y \sim N(0, \sigma_Y^2)\). Suppose \(X_1, \ldots, X_n\) are \(n\) random samples of \(X\) and \(Y_1, \ldots, Y_m\) are \(m\) random samples of \(Y\).
Find the mean and variance of \(\bar{X}_n = \frac{1}{n} \sum_{k=1}^n X_k\).
Verify that \(\bar{X}_n\) is exactly normal for all \(n\). [hint: this is not the central limit theorem.]
Show that \(\bar{X}_n - \bar{Y}_m \sim N\!\left(0,\, \tfrac{\sigma_X^2}{n} + \tfrac{\sigma_Y^2}{m}\right)\).
Suppose \(\sigma_X^2 = 9\) and \(\sigma_Y^2 = 25\). If \(n = 30\) and \(m = 50\), what is the probability that \(\bar{X}_n\) exceeds \(\bar{Y}_m\) by at least \(0.9\)?
Problem 4
Random variables \(X_1, \ldots, X_n\) are independent and identically distributed Cauchy. Identify the distribution of the sample mean \(\bar{X}_n\). Comment on its use as an estimator of the location parameter.
Problem 5
A semiconductor fabrication line produces wafers in batches of 200 chips. Each chip is independently defective with probability 0.04. What is the probability that a batch contains between 6 and 12 defective chips (inclusive)? Compare to the Poisson approximation.
Problem 6
Customer support tickets arrive at a help desk according to a Poisson process with mean 100 tickets per hour. The help desk has staffing to handle up to 120 tickets per hour without overflow. What is the probability that the staff is overwhelmed in any given hour?
Problem 7
Random variables \(U_1, U_2, \ldots, U_{30}\) are independent and uniformly distributed on \([0, 1]\). Compute \(P\!\left[\sum_{k=1}^{30} U_k \le 18\right]\).
Problem 8
A satellite transmitter requires a battery for continuous operation. Each battery has an exponentially-distributed lifetime with mean 50 hours. When a battery fails it is immediately replaced. The satellite stockpile carries 60 batteries. What is the probability that the satellite operates for at least 3,200 hours?
Problem 9
Random variable \(X \sim \chi^2(r)\). Approximate \(P[X > 125]\) for \(r = 100\).
Problem 10
Random samples are drawn from a population with known variance \(\sigma^2 = 49\) and unknown mean \(\mu\). From \(n = 36\) samples the sample mean is \(\bar{X}_n = 71.24\). Construct a 95% confidence interval for \(\mu\).
Problem 11
A flight-delay study estimates the mean delay time per flight at a major airport. Historical data give a standard deviation of 12 minutes. How many flights must the study sample so that a 95% confidence interval for the mean has width at most 4 minutes?
Problem 12
Mary takes 36 random samples from a population with known variance \(\sigma^2 = 49\) and computes the sample mean \(\bar{X}_n = 71.24\). She constructs a 95% confidence interval for the population mean \(\mu\). Suppose the true mean is \(\mu = 73\). Compute the probability that the sample mean of 36 new random samples falls within Mary’s confidence interval.
Problem 13
A statewide ballot measure is supported by 60% of the voting population. A polling firm randomly surveys 100 voters.
What is the probability that between 55 and 70 surveyed voters (inclusive) support the measure?
The firm reports a sample proportion \(\hat{p}_n = 0.62\). Construct a 95% confidence interval for \(p\).
Problem 14
An environmental engineer takes 12 independent measurements of dissolved oxygen in a reservoir. The sample mean is 7.2 mg/L and the sample standard deviation is 0.8 mg/L. The population variance is unknown. Construct a 95% confidence interval for the true mean concentration. [hint, use a t-table.]
Problem 15
A computer system requires a certain component for normal operation. The lifetime of the component is a random variable with mean 100 hours and standard deviation 30 hours. You must immediately replace the component upon failure to preserve normal operation. How many components should you purchase to be 95% confident that the computer can run continuously for 3,000 hours?
Problem 16
Suppose \(X\) and \(Y\) are independent and identically distributed exponential random variables with mean \(\lambda\). Find the pdf of \(Z = X - Y\). Find the characteristic function of \(Z\). Compute \(E[Z]\) and \(V[Z]\).