Homework 7

Problem 0

Daily derivation #7, Derive all the BEG-CUP pdf moments: binomial, geometric, hypergeometric, negative binomial, Poisson, gamma, exponential, chi-square, beta, uniform, Gaussian.

Problem 1

Let \(A\) be an event. Define the indicator function

a. Evaluate \(I_A(x) + I_{A^C}(x)\).

b. Compute \(E[I_A]\).

c. Prove: \(E[X] \ge E[X \cdot I_{\{X \ge c\}}]\) for any constant \(c\) and any positive random variable \(X\) with \(E[X] < \infty\).

Problem 2

Suppose that \(A\) and \(B\) each randomly and independently choose 3 of 10 objects. Find the expected number of objects,

a. chosen by both \(A\) and \(B\).

b. not chosen by either \(A\) or \(B\).

c. chosen by exactly one of \(A\) and \(B\).

Problem 3

Let \(Y = A \, \cos(\omega t) + c\) where \(\omega\) and \(c\) are constants with \(E[A] = \mu\) and \(V[A] = \sigma^2\). Find the mean and variance of \(Y\).

Problem 4

Find the mean and variance of \(X\) if

a. \[f_X(x) = \begin{cases} c (1-x^2) & -1 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}\]

b. \[f_X(x) = \begin{cases} c x (1-x^2) & 0 \le x \le 1 \\ 0 & \textrm{else}. \end{cases}\]

Problem 5

Suppose that \(B\) is a standard uniform random variable, \(B \sim U[0,1]\). Calculate \(E[B^n]\) and \(V[B^n]\).

Problem 6

A dart is equally likely to land at any point inside a circular target of radius 2. Let \(R\) be the distance of the landing point from the origin. Find the mean and variance of \(R\).

Problem 7

Random variable \(X\) equals 2 with probability \(0.4\) and is uniformly distributed \([0, 1]\) otherwise. What is \(E[X] + V[X]\)?

Problem 8

a. Let \(Z \sim N(0, 1)\). Define \(Y = Z + n\) for some constant \(n\). Compute \(E[Y^2]\).

b. Compute \(E[Y^6]\).

Problem 9

Random variable \(X \sim N(0, 4)\). What is \(E[|X|]\)?

Problem 10

Suppose \(Y\) is a non-negative random variable.

a. Show: \(\displaystyle E[Y] = \int_{0}^{\infty} P(Y > t) dt\).

b. Show: \(\displaystyle E[Y^n] = \int_{0}^{\infty} n \, x^{n-1} P(Y > x) dx\).

Problem 11

Random variable \(X_1 \sim N(0, 1)\). Define \(X_i \sim N(X_{i-1}, 1)\). What is the distribution of \(X_n\)?

Hint: \(N(a, \sigma^2) = a + \sigma Z\) where \(Z \sim N(0,1)\).

Problem 12

Let \(X\) be a Pareto random variable, \(f_X(x) = \frac{2}{x^3}\) for \(x \ge 1\), and \(f(x)=0\) otherwise.

a. Compute \(E[X]\).

b. Compute \(E[X^2]\).

Problem 13

Let \(X\) be a Poisson random variable with parameter \(\lambda\). Show directly that \(E[X^n] = \lambda E[( X + 1 )^{n - 1}]\). Use the result to compute \(E[X^3]\).

Problem 14

Suppose that \(X\) is a random variable with pdf

Find \(a\) and \(b\) if \(E[X] = \frac{3}{5}\).

Problem 15

Consider the following limiter, \(g(x)\):

a. Find expressions for the mean and variance of \(Y = g(X)\) for an arbitrary continuous random variable \(X\) with symmetric pdf, \(f_X(x)\).

b. Suppose \(a = 1\). Evaluate the mean and variance if \(X\) is a Laplacian random variable with \(\lambda = 1\), that is \(f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)\).

c. Suppose \(a = \frac12\). Evaluate the mean and variance if \(X = U^3\) where \(U \sim U[-1, 1]\).

Problem 16

Consider the following limiter with center-level clipping, \(h(x)\):

a. Find expressions for the mean and variance of \(Y = h(X)\) for an arbitrary continuous random variable \(X\) with symmetric pdf, \(f_X(-x) = f_X(x)\).

b. Suppose \(a = 1\) and \(b = 2\). Evaluate the mean and variance if \(X\) is a Laplacian random variable with \(\lambda = 1\), that is \(f_X(x) = \frac{1}{2 \lambda} \exp\left(-\frac{|x|}{\lambda}\right)\).

c. Suppose \(a = \frac12\) and \(b = \frac34\). Evaluate the mean and variance if \(X = b \cos(2 \pi U)\) where \(U \sim U[-1, 1]\).