Question

Suppose that for two random variables X and Y the joint density function is

f(x,y)=4xe^(−x(y+4))
for x>0 and y>0. Find each of the following:
(a) fX|Y(x,y)=
(b) fY|X(x,y)=

Answer 1

To find the conditional probability density function (pdf) of X given Y, we need to use the joint density function and the marginal pdf of Y. We can then substitute these into the formula to find fX|Y(x,y).

Step-by-step explanation:

Joint Density Function

To find the conditional probability density function (pdf) of X given Y, we need to use the formula:

fX|Y(x,y) = f(x,y) / fY(y)

In this case, f(x,y) = 4xe^(-x(y+4)). The marginal pdf of Y, fY(y), can be found by integrating f(x,y) with respect to x, from 0 to infinity. Once we have fY(y), we can substitute it into the formula to find fX|Y(x,y).

Solution

To find fY(y), we integrate f(x,y) with respect to x:

fY(y) = ∫(from 0 to infinity) 4xe^(-x(y+4)) dx

This integral can be evaluated using integration techniques. Once we have fY(y), we substitute it into the formula to find fX|Y(x,y).