Finding the marginal density function when given a joint density function.

Question

So I have gotten the following joint density function for random continuous variables $X$ and $Y$.

$$ f_{X,Y}(x,y) = \begin{cases} e^{-y} & \mbox{if } 0 < x < y <\infty \\ 0 & \mbox{otherwise.} \end{cases} $$

I want to find the marginal density functions of X. I tried doing the following; $$ f_X(x) = \int_0^\infty e^{-y}dy = 1 $$ Because $e^{-y}$ is a exponential distribution with parameter $\lambda = 1$. However, the answer should be $f_X(x) = e^{-x}$.

I suspect that I am not properly taking the boundaries in account. How can I do that correctly?

Thanks for reading,

K.

Answer 1

Hint: You know that $x < y <\infty$. Thus the lower bound for $y$ is $x$ and the upper bound is $ \infty$. $$ f_X(x) = \int_x^\infty e^{-y}dy $$

Then the range for $x$ at $f_X(x)$ is $0<x<\infty$