Problem 1

Say X be a random variable with \(P(X=0)=P(X=1)=1/2\), and Y a random variable with conditional density \(f_{Y|X=x}(y|x) = cy^x(1-y);0<y<1\). Find the mean and variance of Y using theorem 1.9.6.

Problem 2

Let X be a random variable with moment generating function \(\psi\). Assuming the conditions of theorem 1.10.5 hold show that

\[\frac{d^2 \log \psi(t) }{dt^2}|_{t=0}=var(X)\]

Use this to find the variance of \(X\sim Geom(p)\)

Problem 3

Let X be a random variable with density \(f(x)=3x^2\), \(0<x<1\). Find the moment generating function of X.

Problem 4

  1. Find a condition on the moment generating function of a random variable X so that X and Y=-X have the same distribution. Give an example.

  2. Find a condition on the moment generating function of a random variable X so that X and Y=1-X have the same distribution. Give an example.