Dividing probability density functions?
I have a problem and I am at a loss:
Suppose that Z is a standard normal random variable and Y1 and Y2 are χ2 distributed random variable with υ1 and υ2 degrees of freedom, respectively. Further, assume that Z and Y1 and Y2 are independent.
a) Define W = (Z/SQRT(Y1)). Find E(W) and V(W). What assumptions do you need about the value of υ1?
b) Define U= Y1/Y2 . Find E(U) and V(U). What assumptions do you need about the value of υ1 and υ2?
I know the moment generating functions of Z, Y1 & Y2, but I don't know how I am supposed to find the distribution of W considering we are dividing densities... Any help would be vastly appreciated!