Boundaries of the marginal density from a joint distribution

Question

I know that to find the marginal density I need to compute

$f(x) = \int_{-\infty}^{\infty}f(x,y)dy$

But when I have i.e.

$f(x,y) = (1/θ^2)e^{-y/θ}$ for $0<x<y<\infty$

Then to my understanding it is calculated

$f(x) = \int_{0}^{x}(1/θ^2)e^{-y/θ}dy$

$f(y) = \int_{0}^{y}(1/θ^2)e^{-y/θ}dx$

Yet on some resources I find the integral going from i.e. $\int_{x}^{\infty}$

What is the general rule for the boundaries?

Answer 1

If $f(x,y)$ is density function concentrated on $x>0,y>0$ then the first marginal is $f(x)=\int_0^{\infty} f(x,y)dy$. In case $f(x,y)=0$ for $y <x$ this reduces to $f(x)=\int_x^{\infty} f(x,y)dy$ because the integral from $0%=$ to $x$ vanishes.