Marginal and Joint Distribution

Question

Find the marginal probability density function of x.

I found the marginal of y but cannot understand how to find the marginal of x. Can someone explain how do we approach this problem?

Answer 1

For $x \geq 0$, $f(x,y)$ is defined for $y > x$. Hence $$f(x) = \int_{y=x}^{\infty}f(x,y) \mathrm{d} x=\int_{t=0}^{\infty} e^{-x}\frac{t e^{-t}}{4}\mathrm{d}t, $$ which can be solved by integration by parts.

For $x \leq 0$, $f(x,y)$ is defined for $y >-x$, which can be solved then similarly.