Question

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.)

fZ(z)=∫∞0fY,Z(y,z)dy

fZ(z)=∫∞0fY,Z(y,z)dz

fZ(z)=∫∞0fY(y)fZ,Y(z,y)dy

fZ(z)=∫∞0fY(y)fZ|Y(z|y)dy

fZ(z)=∫∞0fY(y)fX(yz)dy

fZ(z)=∫∞0yfY(y)fX(yz)dy

Answer 1