Finding the value of a constant in a joint probability density function

Question

Given a joint density function $f(x, y) =Ae^{-x-y}$, where $0 < x < y < \infty$, find the value of the constant $A$.

To find the value of $A$, clearly I need to take the double integral, however I'm struggling to work out what the limits should be. What should they be?

Answer 1

$\int_{0}^{\infty}\int_{x}^{\infty} {Ae^{-x-y}}dydx=\int_{0}^{\infty}Ae^{-x}\int_{x}^{\infty}e^{-y}dydx=\int_{0}^{\infty}Ae^{-2x}dx={A\over2}=1\to A=2$

Answer 2

We have the following situation as equivalent to $$\int_{0}^{\infty} \int_{0}^{y} Ae^{-x}e^{-y}\, dx \, dy = 1 $$ $$\int_{0}^{\infty} Ae^{-y}\left (1-e^{-y}\right)\, dy =1$$ $$\implies A - \frac {A}{2} =1 $$ $$\implies A=2$$