Find c such that f(x,y) is a Joint Probability Density Function

Question

The joint probability function of X and Y is given by $$f(x,y) =\left\{ ce^{(-x-3y)}: 0 <x<\infty, 0<y<\infty \right\}\\0 \ \ \ \ \ \ otherwise $$

a) For what value of c is this a joint density function?

Answer 1

In order to be a pdf, $f$ needs to be integrable to 1:

$1= \int_0^\infty \int_0^\infty ce^{-x-3y}dxdy = c \int_0^\infty e^{-x} dx \int_0^\infty e^{-3y}dy = c \left(-e^{-x}|_0^\infty\right) \left(-e^{-3y}/3|_0^\infty\right) = c \cdot 1 \cdot \frac{1}{3}, $ so $c=3$.