Let the rv Z be the standard normal Show that Y Z2 has the

Let the r.v. Z be the standard normal. Show that Y = Z^2 has the Chi-square X^2 1 distribution. That is, show that Y has the density function fY(y) = 1 / squereroot2 pi 1 / squareroot y e^v/2. Let X and Y be independent random variables that X ~ Gamma(alpha1, beta) and Y ~ Gamma(alpha2, beta). Show that X + Y ~ Gamma(alpha1 + alpha2, beta).

Solution