Probability help please?

Question

Suppose X and Y have joint density f(x,y) = e^(-x-y) for x,y > 0.

What are the marginal distributions of X and Y?

Answer 1

the marginal for x is what you get when you "sum up" (integrate) over all values of y:

g(x)= Integral from 0 to infinity of f(x,y) dy

Treat x as a constant when performing the integration.

Note that the integrand simplifies to the product of an exponential containing x alone times an exponential containing y alone, so the integration is easy.