technique to find the density function for the random variable Y. The density function should be given as a piecewise function.   f(x) = 1/3e^-1/3x      x>0  Y=e^x     0 elsewhere

Example: The Gamma Distribution Suppose X ~ T(a, B), then M(t) = dx BaT(a) 22 Now we do a little high school algebra. Define & = Then one may write 1-Bt" Equation (22) as: M(t) = ()ª / (23) dx One recognizes the integrand of Equation (23) as a r(a, 6) density, hence 8. M(t) = (Gr"(1) = (1 - AY• (24) (1 – Bt)a Exercise 5. Verify the algebra leading from Equation (22) to Equation (23).

The joint density function for random variables X and Y is f(x, y). Find the conditional mean and variance of X given Y = 1/2      f(x,y) = 12/7(x^2 + xy)     0<x<1, 0<y<1      0 elsewhere     = _mean1/2________________       = _var1/2________________

Let X and Y be random variables with joint density function ,0 < x < 4 and 1 < y < 5 f(x,y) = {xy/96 | ,otherwise. } Find the joint density function g(u,v) for the transformed variables U = X + 2Y and V = X. Specify the domain of g and draw the domain.