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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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    Please not this is just intro to probability, not the more advanced course with measure theory2012-12-19
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    Have a look at [this](http://math.stackexchange.com/questions/158206/expression-for-n-th-moment) for $n=1$.2012-12-19
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    I answered that question in great detail here http://math.stackexchange.com/questions/255937/problems-on-expected-value/256006#2560062012-12-19
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    @StefanHansen - I believe that post uses what I want to prove2012-12-19
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    @Learner - I looked at tht post at the link, but I didn't understand which one of the questions asked by the OP is my question. I also looked at your answer and didn't manage to understand which part is relevant to this question2012-12-19
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    Belgi: Fubini? $ $2012-12-19
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    @Belgi Take a special case of my answer. $X^- = 0$ a.s. and $p=1$, that gives the answer to question 1 (2 follows also from the Markov inequality in my answer there.2012-12-19
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    Belgi, the proof was however discussed in the comments. It leads to [this](http://math.stackexchange.com/a/64199/25632) for example.2012-12-19
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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

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