Chapter 6.4, Problem 34E

A density function sometimes used by engineers to model lengths of life of electronic components is the Rayleigh density, given by

$f(y)=\{\begin{array}{ll}\left(\frac{2y}{\theta }\right)e^{-y^2/\theta },\hfill & y>0,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

1. a If Y has the Rayleigh density, find the probability density function for U = Y².
2. b Use the result of part (a) to find E(Y) and V(Y).