Dividing probability density functions?

I have a problem and I am at a loss:

Suppose that Z is a standard normal random variable and Y**1** 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(Y**1**)). Find E(W) and V(W). What assumptions do you need about the value of υ1?

b) Define U= Y**1**/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!