Question

Let f(x) = xe−x if x ≥ 0 and f(x) = 0 if x < 0 Is f a probability density function?

Answer 1

Yes. $f(x)$ is a PDF if

• $f(x) \ge 0$ for all $x$ in the support; this is true, since $e^{-x}>0$ for all $x$, and multiplying a positive number $x$ doesn't change this fact;

• the integral of $f(x)$ over its support is 1; this is also true, since

$\displaystyle \int_{-\infty}^\infty f(x) \, dx = \int_0^\infty xe^{-x} \, dx = 1$

which can be computed by parts.