Homework #9

EE 364: Spring 2026

ImportantAssignment Details

Assigned: 02 April
Due: Thursday, 09 April at 16:00

BrightSpace Assignment: Homework 9

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 #9

  • State the formal definition for all UC-MOPED convergences.
  • Prove the Weak Law of Large Numbers: \(\bar{X}_n \xrightarrow{p} \mu_X\).

Problem 1

Random variable \(X\) is exponential with parameter \(\lambda = 1\). Given \(X = x\), the random variable \(Y\) is uniformly distributed on \([0, x]\).

  1. Write the conditional pdf \(f_{Y|X}(y \mid x)\) and find the joint pdf \(f_{X,Y}(x, y)\).

  2. Sketch the support of \(f_{X,Y}\).

  3. Find the marginal pdf \(f_Y(y)\).

Problem 2

Let \(X\) and \(Y\) have joint pdf \(f_{X,Y}(x, y) = k \, e^{-3x - 4y}\) for \(0 < y < x\).

  1. Find \(k\).

  2. Find the conditional pdf \(f_{X|Y}(x \mid y)\).

  3. Compute \(P\!\left[X > \tfrac{5}{2} \mid Y = 2\right]\).

Problem 3

  1. Find the correlation coefficient \(\rho(X, X)\).

  2. Find the correlation coefficient \(\rho(X, cX)\) for constant \(c \ne 0\).

Problem 4

Let \(X_1, \ldots, X_n\) be i.i.d. with mean \(\mu\) and variance \(\sigma^2\). Consider the estimator \(\hat{\theta}_n = (\bar{X}_n)^2\) for \(\mu^2\). Find \(E[\hat{\theta}_n]\) and \(V[\hat{\theta}_n]\).

Problem 5

Let \(X_1, \ldots, X_n\) be i.i.d. \(U[0, a]\) where \(a\) is unknown. Define \(Y_n = \max(X_1, \ldots, X_n)\).

  1. Find \(E[Y_n]\).

  2. Find \(V[Y_n]\).

Problem 6*

  1. Prove the Cauchy-Schwarz inequality: for \(a_k, b_k \in \mathbb{R}\), \[\left(\sum_{k=1}^{n} a_k b_k\right)^2 \le \left(\sum_{k=1}^{n} a_k^2\right) \left(\sum_{k=1}^{n} b_k^2\right)\]

  2. Let \(X_1, \ldots, X_n\) be i.i.d. with mean \(\mu\) and variance \(\sigma^2\). Define \(\hat{\theta}_n = \sum_{k=1}^{n} c_k X_k\) where \(\sum_{k=1}^{n} c_k = 1\). Find \(E[\hat{\theta}_n]\) and \(V[\hat{\theta}_n]\) in terms of the \(c_k\).

  3. Show that \(V[\hat{\theta}_n] \ge \sigma^2/n\) and that \(c_k = 1/n\) achieves equality.

Problem 7

  1. Let \(g\) be a non-negative function and \(c > 0\). Prove that \(P[g(X) \geq c] \leq \frac{E[g(X)]}{c}\).

  2. Show that \(P[X \geq a] \leq e^{-ta} \, E[e^{tX}]\) for any \(t > 0\).

  3. Let \(X \sim \textrm{Exp}(\lambda)\). Compute \(E[e^{tX}]\) for \(t < 1/\lambda\). Find the tightest upper bound on \(P[X \geq a]\) for \(a > \lambda\) by minimizing over \(t\). Compare to the exact value \(P[X \geq a]\).

Problem 8

Suppose that the number of particle emissions by a radioactive mass in \(t\) seconds is a Poisson random variable with mean \(\lambda t\). Use the Chebyshev inequality to obtain a bound for the probability that \(\left|N(t)/t - \lambda\right|\) exceeds \(\epsilon\).

Problem 9

A fair die is tossed 20 times. Bound the probability that the total number of dots is between 60 and 80.

Problem 10

Let \(\zeta \sim U[0,1]\). Define the following sequences of random variables for \(n \ge 1\): \[X_n(\zeta) = \zeta^n \qquad Y_n(\zeta) = \cos^2 2 \pi \zeta \qquad Z_n(\zeta) = \cos^n 2 \pi \zeta\] Do the sequences converge, and if so, in what sense and to what limiting random variable?

Problem 11

Let \(X_n \sim \textrm{Cauchy}\!\left(\frac{1}{n}\right)\). Show that \(X_n\) converges in probability to zero.

Problem 12

Let \(U \sim \textrm{Uniform}(0,1)\) and define \[X_n = n \, I_{\left[0,\, 1/\sqrt{n}\right]}(U), \qquad n = 1, 2, 3, \ldots\]

  1. Does \(X_n\) converge in probability to zero?

  2. Does \(X_n\) converge in mean square to zero?

Problem 13

Let \(X_n\) be a sequence of Laplacian random variables with parameter \(\alpha = n\). Does this sequence converge in distribution? [Hint: compute \(F_{X_n}(x)\) and evaluate \(\lim_{n \to \infty} F_{X_n}(x)\) for \(x < 0\), \(x = 0\), and \(x > 0\).]