2

Possible Duplicate:
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 ?

Community
  • 1
Belgi
  • 23,150
  • Please not this is just intro to probability, not the more advanced course with measure theory – Belgi Dec 19 '12 at 13:45
  • Have a look at this for $n=1$. – Stefan Hansen Dec 19 '12 at 13:48
  • I answered that question in great detail here http://math.stackexchange.com/questions/255937/problems-on-expected-value/256006#256006 – Learner Dec 19 '12 at 13:48
  • @StefanHansen - I believe that post uses what I want to prove – Belgi Dec 19 '12 at 13:51
  • @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 question – Belgi Dec 19 '12 at 13:52
  • Belgi: Fubini? $ $ – Did Dec 19 '12 at 13:54
  • @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. – Learner Dec 19 '12 at 13:55
  • Belgi, the proof was however discussed in the comments. It leads to this for example. – Stefan Hansen Dec 19 '12 at 14:10
  • @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. – Belgi Dec 19 '12 at 14:14

0 Answers0