Homework 2

Published

February 15, 2026

Homework 2 is due 2/22/2026 at 11:59 PM. Submit your homework on Canvas as one PDF document.

  1. Show that the moment generating function for a random variable \(X\sim Gamma(\alpha,\beta)\) is \(\left(1-\beta t\right)^{-\alpha}\).

  2. Let \(X\) be a discrete random variable with PMF

\[ P(X=x)=\left\{\begin{array}{cc} 0.25+0.5e^{-\theta} & x =0 \\ 0.25+0.5\theta e^{-\theta} & x =1 \\ 0.5\frac{\theta^x}{x!}e^{-\theta} & x = 2,3,\ldots\\ 0 & \textrm{otherwise} \end{array}\right. \] Show that the PMF is valid.

  1. Find the moment-generating function of \(X\sim\chi^2_k\) and \(f_X(x)=\frac{x^{k/2-1}\exp\{-x/2\}}{2^{k/2}\Gamma(k/2)}\) with \(x>0\).

  2. Problem 110 in Chapter 4 of Modern Mathematical Statistics (2021; 3rd Edition)

  3. Problem 113 in Chapter 4 of Modern Mathematical Statistics (2021; 3rd Edition)