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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?

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You are looking for all the continuous random variables which are supported in $(-1;+\infty)$, and with expectation $0$. For example, a uniform law on $(-a,a)$, $0<a<1$ works. – Davide Giraudo Apr 17 '12 at 16:32
Necessarily, $\lim_{x\to\infty}f(x)=0$, if that helps. – akkkk Apr 17 '12 at 16:34

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