91Ó°ÊÓ

Suppose \(\mathrm{X}\) has the Poisson distribution with parameter \(\lambda\). Let \(\mathrm{Y}=1 \quad\) if \(\mathrm{X}\) is even \(-1 \quad\) if \(\mathrm{X}\) is odd. Find the distribution of \(\mathrm{Y}\). Find the distribution of \(\mathrm{Y}\).

Suppose \(\mathrm{X}\) has a normal distribution with mean 0 and variance 1 . Let \(\mathrm{Y}=\mathrm{X}^{2}\). Find the distribution of \(\mathrm{Y}\) using the moment generating function technique.

Two points, \(\mathrm{A}\) and \(\mathrm{B}\), are chosen on the circumference of a circle so that the angle at the center is uniformly distributed over \((0, \pi)\). What is the probability that the length of \(\mathrm{AB}\) exceeds the radius of the circle?

Suppose X has density function: $$ f(X)=\left[\pi\left(1+X^{2}\right)\right]^{-1} \quad-\infty