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Problems on expected value

I have the following exercise I wish to solve:

Let $X$ be a continuous random variable with distribution function $F$ and continuous density function $f$.

Assume that $X$ is nonnegative; that is, $\forall x\leq0\, F(x)=0$.

  1. Assume that $Ex<\infty$. Prove that $EX=\int_{0}^{\infty}(1-F(x))\, dx$

  2. Assume that $Ex=\infty$. Prove that $\int_{0}^{\infty}(1-F(x))\, dx=\infty$

What I tried:

Using integration by parts with $u=1-F(x),v'=1$ (hence $u'=-f(x),v=x)$ we get $\int_{0}^{\infty}(1-F(x))\, dx=(1-F(x))x|_{0}^{\infty}-\int_{0}^{\infty}-f(x)x\, dx$

$=(x-xF(x))|_{0}^{\infty}+\int_{0}^{\infty}f(x)x\, dx$

$=(x-xF(x))|_{0}^{\infty}+EX$

Since $(x-xF(x))|_{0}=0$

we need to prove $\lim_{x\to\infty}x(1-F(x))=0$ which I tried to show by L'Hôpital's rule and failed.

Can someone please help me continue with my way, or suggest another way to prove the requested equality ?

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    @StefanHansen - did's answer is way more advanced than what I know right now...the probability course I am taking is not about mwasure theory and is more basic.2012-12-19

0 Answers 0