Homework #8

EE 364: Spring 2026

ImportantAssignment Details

Assigned: 26 March
Due: Thursday, 02 April at 16:00

BrightSpace Assignment: Homework 8

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 #8

  • Derive the pdf for \(Y = g(X)\) if \(g(x)\) is 1-to-1.
  • Prove the population Uncertainty Principle: \(\sigma_{XY}^2 \le \sigma_X^2 \sigma_Y^2\).

Problem 1

Let \(A \times B = \{(x, y) : x \in A \textrm{ and } y \in B\}\). Suppose \(A = \{ a_1, a_2 \}\) and \(B = \{ b_1, b_2, b_3 \}\).

  1. What is the Cartesian product \(A \times B\)? How many elements are in \(2^{A \times B}\)? Produce four sub-collections \(\mathcal{A} \subset 2^{A \times B}\) that are sigma-algebras.

  2. Let \(A\), \(B\), \(C\), \(X\), and \(Y\) be sets. Then prove or disprove:

    1. If \(A \subset X\) and \(B \subset Y\) then \(A \times B \subset X \times Y\).

    2. \((A \cup B) \times C = (A \times C) \cup (B \times C)\).

Problem 2

Random variable \(Y = e^X\). Find the pdf of \(Y\) when \(X\) is a Gaussian random variable – in this case \(Y\) is said to be a lognormal random variable. Plot the pdf and cdf of \(Y\) when \(X\) is zero-mean with variance \(\frac{1}{8}\). Repeat with variance 8.

Problem 3

Let \(Y = \alpha \tan \left( \pi X \right)\), where \(X\) is uniformly distributed in the interval \([-1, 1]\). Determine the pdf \(f_Y(y)\) and identify \(Y\) as one of the BEG-CUP random variables.

Problem 4

Let a radius be given by the random variable X as: \[f_X(x) = \begin{cases} c x \left( 1 - x^2\right) & 0 \le x \le 1 \\ 0 & \textrm{else} \end{cases}\]

  1. Find the pdf of the area covered by a disc with radius \(X\).

  2. Find the pdf of the volume of a sphere with radius \(X\).

Problem 5

A voltage \(X\) is a Gaussian random variable with mean 1 and variance 2. Find the pdf of the power dissipated by an \(R\)-\(\Omega\) resistor \(P = R X^2\).

Problem 6

For the pair of random variables \((X, Y)\), sketch and shade the region of the plane corresponding to the following events. For each, write the probability as a double integral with explicit bounds (some may require more than one integral). You do not need to evaluate the integrals.

  1. \(\{X + Y > 3\}\)

  2. \(\{|X - Y| \ge 1\}\)

  3. \(\{XY < 0\}\)

  4. \(\{\max(|X|, Y) < 3\}\)

Problem 7

Let \(X\) and \(Y\) have joint pdf \(f_{X,Y}(x, y) = k(x + y)\) for \(0 \le x \le 1\), \(0 \le y \le 1\).

  1. Find \(k\).

  2. Find the marginal pdfs \(f_X(x)\) and \(f_Y(y)\).

  3. Find \(P[X < Y]\), \(P[Y < X^2]\), and \(P[X + Y > 0.5]\).

Problem 8

Let \(X\) and \(Y\) have joint pdf \(f_{X,Y}(x, y) = 2 e^{-x - y}\) for \(0 < x < y\).

  1. Sketch the support region.

  2. Find the marginal pdfs \(f_X(x)\) and \(f_Y(y)\).

  3. Are \(X\) and \(Y\) independent?

  4. Compute \(\textrm{Cov}(X, Y)\).

Problem 9

Let \(N_1\) be the number of web page requests arriving at a server in a 100-ms period and let \(N_2\) be the number of page requests arriving at a server in the next 100-ms period. Assume that in a 1-ms interval either zero or one page request takes place with respective probabilities \(1 - p = 0.95\) and \(p = 0.05\), and that requests in different 1-ms intervals are independent of each other.

  1. Describe the underlying sample space \(S\) and show the mapping from \(S\) to \(S_{N_1, N_2}\), the range of the pair \((N_1, N_2)\).

  2. Find the joint pmf of \(N_1\) and \(N_2\).

  3. Find the marginal pmf for \(N_1\) and for \(N_2\).

  4. Find the probability of the following events: \(A = \{N_1 \ge N_2\}\), \(B = \{N_1 = N_2 = 0\}\), \(C = \{N_1 > 5, N_2 > 3\}\), \(D = \{N_1 + N_2 = 10\}\).

Problem 10

Random variables \(X\) and \(Y\) have joint probability density \(f(x,y) = \frac{1}{2}(x^2 + \sqrt{x}\, y)\) on the closed rectangle \([0, 1] \times [0, 2]\). Are the random variables positively correlated?

Problem 11

Let \(X \sim U[-a, a]\) and \(Y = X^2\). Prove or disprove: \(X\) and \(Y\) are uncorrelated.

Problem 12

Suppose that random variable \(Y\) is standard normal. Suppose that random variable \(X\) is also zero-mean Gaussian but it has six times the variance that \(Y\) has. Suppose further that \(X\) and \(Y\) are correlated with coefficient \(\rho = 0.5\). Is that enough information to compute the variance of the random variable \(Z = 2X - Y\)? If so, find it.

Problem 13

Random variable \(X\) has population standard deviation \(\sqrt{V[X]} = 4\) while random variable \(Y\) has population variance \(V[Y] = \frac{1}{9}\). They have population linear correlation coefficient \(\rho_{XY} = -\frac{1}{2}\). Find the population covariance \(\textrm{Cov}(2X - 1,\; 3Y + 2)\).

Problem 14*

  1. A positive random variable \(X\) takes value 2 with probability \(p\) and value 6 with probability \(1 - p\). Compute \(E[X] \cdot E[1/X]\) as a function of \(p\) and verify that \(E[X] \cdot E[1/X] \ge 1\) for all \(p \in [0, 1]\).

  2. Let \(X \sim \chi^2(r)\) for \(r > 2\). Compute \(E[X] \cdot E[1/X]\).

  3. Show that \(E[X] \cdot E[1/X] \ge 1\) for any positive random variable \(X > 0\).