# Find the value of $c$ for a joint probability density function.

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $$f(x,y)=\left\{ \begin{array}{lcc} ce^{-x}& \text{if } x \geq 0, |y|< x; \\ \\ 0 & \text{otherwise.} \end{array} \right.$$ Find $c$ such that $f$ is a joint probability density function.

My work

$f$ is a density function if $f(x,y)\geq 0$ for all $x,y\in\mathbb{R}$ and $\int\int f(x,y)dydx=1$

Then,

$1=\int_R\int_R f(x,y)dydx=\int_R\int_R ce^{-x}dydx=c\int_R\int_R e^{-x}dydx$

I have problem finding the area of integration. Can someone help me with this?

Note that $f(x,y)$ is not zero for $-x < y < x$ and $x\ge 0$. Therefore, $$c \int_{0}^{\infty} \int_{-x}^{x} e^{-x} dy dx = 2c \int_{0}^{\infty}xe^{-x} dx = 1.$$
• Thanks, i follow your hint and i take $c=1/2$ – Bvss12 Jul 13 '18 at 22:07