(Solved): Consider X and Y are continuous random variables having joint probability density function: f(x,y)= ...

Expert Answer

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   To find the conditional probability density function of Y given X=2, we can use the formula:
f(y|x) = f(x,y) / f(x)

where f(x,y) is the joint probability density function and f(x) is the marginal probability density function of X.

To find f(x), we integrate the joint density function over all possible values of Y:

f(x) = ? f(x,y) dy
= ? 4xe^(-x-y) dy (from y=0 to y=?)
= 4x e^(-x) (integration of e^(-x-y) with respect to y)

To find f(y|x=2), we substitute x=2 into the joint density function and divide by the marginal density function evaluated at x=2:

f(y|x=2) = f(2,y) / f(2) = 4(2)e^(-2-y) / (4(2)e^(-2)) = e^(-y-2) / 2



Please refer to solution in this step.