Homework 2

Published

September 6, 2022

Homework 2 is due 9/16/2022 at 11:59 PM. Submit your homework on Canvas as one PDF document.

The PDF version of this assignment can be found here.

  1. X is a random variable with the following pdf f(x)=\frac{2x}{\pi^2} for x\in [0,\pi], show that the pdf is valid.

  2. X\sim Pois(\lambda), what is the moment generating function of Y=\log(X+4)?

  3. 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.

  4. Let Y be random variable with support (0,1). What is the normalizing constant c to make f(x)=cx valid?