Let X and Y be independent positive random variables. Let Z=X/Y. In what follows, all occurrences of x, y, z are assumed to be positive numbers.

(1) Suppose that X and Y are discrete, with known PMFs, pX and pY. Then,

pZ|Y(z|y)=pX(?).

What is the argument in the place of the question mark?

(2) Suppose that X and Y are continuous, with known PDFs, fX and fY. Provide a formula, analogous to the one in part (a), for fZ|Y(z|y) in terms of fX. That is, find A and B in the formula below.

fZ|Y(z|y)=AfX(B).

A=

B=

(3) Which of the following is a formula for fZ(z)?

fZ(z)=

(Choose all that apply.)

A - fZ(z)=0fY,Z(y,z)dy

B - fZ(z)=0fY,Z(y,z)dz

C - fZ(z)=0fY(y)fZ,Y(z,y)dy

D - fZ(z)=0fY(y)fZ|Y(z|y)dy

E - fZ(z)=0fY(y)fX(yz)dy

F - fZ(z)=0yfY(y)fX(yz)dy