1. ## Properties of Expecation

Hi,

I have a problem in understanding the following property of expecation:

see attached

Why does E(X) only exist if this condition holds?

Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

Thanks

2. ## Re: Properties of Expecation

Originally Posted by Marmy
Hi,

I have a problem in understanding the following property of expecation:

see attached

Why does E(X) only exist if this condition holds?

Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

Thanks
$E(X)=\int_{-\infty}^{\infty} xf(x) dx$

which is an improper integral, and means:

$E(X)=\lim_{A,B \to \infty}\int_{-A}^{B} xf(x) dx=\lim_{A\to \infty}\int_{-A}^{0} xf(x) dx + \lim_{B \to \infty}\int_{0}^{B} xf(x) dx$

So the expectation exist iff the two limits on the right exist.

Now try integration by parts.

CB

3. ## Re: Properties of Expecation

Originally Posted by Marmy

Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

Thanks
No, $F(x)$ is the cumulative distribution, integral of $F(X)$ from $- \infty$ to $+ \infty$ should is infinite.

$F(x)=\int_{-\infty}^x f(\xi)\; d\xi$

CB

4. ## Re: Properties of Expecation

Thanks! Great help!