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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?

  • 0
    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 works.2012-04-17
  • 0
    Necessarily, $\lim_{x\to\infty}f(x)=0$, if that helps.2012-04-17

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