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.