I am looking for the family of distributions that satisfy the following condition:
$$\int_{-1}^{+\infty}f(x)x d x=0$$
and with this other conditions on $f(x)$:
$$f(x)\ge 0 \text{ in }(-1,+\infty]$$
$$\int_{-1}^{+\infty}f(x) d x =1$$
Any idea on how to solve it?