91Ó°ÊÓ

Let \(X\) have the probability distribution. Show that the random variable \(Y=-2 \ln X\) has a chi-squared distribution with 2 degrees of freedom.

A dealer's profit, in units of \(\$ 5000,\) on a new automobile is given by \(Y=X^{2},\) where \(X\) is a random variable having the density function $$f(x)=\left\\{\begin{array}{ll}2(1-x), & 0 < x<1 \\ 0, & \text {}\end{array}\right.$$ (a.) Find the probability density function of the random variable \(Y\) (b) Using the density function of \(Y,\) find the probability that the profit will be less than \(\mathrm{S} 500\) on the next new automobile sold by this dealership.

A random variable \(X\) has the geometric distribution \(g(x ; p)=p q^{x-1}\) for \(x=1,2,3 .\) Show that the moment-generating function of \(X\) is, and then use \(M x(t)\) to find the mean and variance of the geometric distribution.