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 $\int\int f(x,y)dydx=1$


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

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, $$c \int_{0}^{\infty} \int_{-x}^{x} e^{-x} dy dx = 2c \int_{0}^{\infty}xe^{-x} dx = 1.$$

| cite | improve this answer | |
  • $\begingroup$ Thanks, i follow your hint and i take $c=1/2$ $\endgroup$ – Bvss12 Jul 13 '18 at 22:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.