Properties of Expecation

**Marmy** · January 6th, 2012, 07:14 AM

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

**CaptainBlack** · January 6th, 2012, 09:05 AM

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

**CaptainBlack** · January 6th, 2012, 09:07 AM

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

**Marmy** · January 7th, 2012, 06:58 AM

Thanks! Great help!

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