How do you show $\displaystyle E(X) = \int_{0}^{\infty} P(X > \lambda) \ d \lambda $? We know that $\displaystyle E(X) = \int_{0}^{\infty} \lambda \ p(\lambda) \ d \lambda $. In other words, $\displaystyle P(X > \lambda) $ controls all the moments.

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- Jun 19th 2010, 12:07 PMSamprasprobability
How do you show $\displaystyle E(X) = \int_{0}^{\infty} P(X > \lambda) \ d \lambda $? We know that $\displaystyle E(X) = \int_{0}^{\infty} \lambda \ p(\lambda) \ d \lambda $. In other words, $\displaystyle P(X > \lambda) $ controls all the moments.

- Jun 20th 2010, 06:09 AMtheodds
Try integration by parts, noting that $\displaystyle P(X > \lambda) = 1 - F(\lambda)$ where F is the CDF. Also, I think you are missing an assumption on the support of $\displaystyle X$.

- Jun 20th 2010, 03:03 PMjamix
- Jun 21st 2010, 04:21 PMSampras
$\displaystyle X $ is non-negative.

So we know that $\displaystyle E(X) = \int_{0}^{\infty} \lambda p(\lambda) \ d \lambda $. Let $\displaystyle u = \lambda $ and $\displaystyle dv = p(\lambda) \ d \lambda $. Then $\displaystyle du = 1 $ and $\displaystyle v = P(X \leq \lambda) $. So

$\displaystyle E(X) = \lambda P(X \leq \lambda)- P(X \leq \lambda) $

$\displaystyle = \lambda P(X \leq \lambda)- E(X) = \int_{0}^{\infty} P(X \leq \lambda) \ d \lambda $

$\displaystyle = \lambda P(X \leq \lambda)- E(X) = \int_{0}^{\infty} 1-P(X > \lambda) \ d \lambda $

$\displaystyle = \lambda P(X \leq \lambda)- E(X) = \lambda- \int_{0}^{\infty} P(X > \lambda) \ d \lambda $

So we have:

$\displaystyle \int_{0}^{\infty} P(X > \lambda) \ d \lambda = E(X)-\lambda [\underbrace{P(X \leq \lambda)-1}_{0}] $

and so the result follows. Is this correct?

- Jun 21st 2010, 04:27 PMSampras
Or more generally, one can show that $\displaystyle E(X^p) = p \int_{0}^{\infty} \lambda^{p-1} P(X > \lambda) \ d \lambda $ using induction on $\displaystyle p $? One could use Markov's inequality: $\displaystyle P(X \geq \lambda) \leq \frac{E(X)}{\lambda} $?

- Jun 22nd 2010, 09:55 AMjamix
Unfortunately integrating by parts won't help much here because were evaluating a definite integral. Your first line would actually be this:

$\displaystyle E(X) = \infty - \int_{0}^{\infty} P(X \leq \lambda) $

which is obviously no good.

I don't know the answer off the top of my head. I'll post back if I have any useful hints. - Jun 22nd 2010, 01:58 PMawkward