Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

expected value of a pdf

If we're given a probability density function

,

,

Why is it when we calculate the expected value, we get an undefined expression:

When obviously from the graph, it should be at x = 0?

Also, for an even graph (symmetric about the y-axis), should

all be at x = 0?

Thanks

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: expected value of a pdf

Hi identity;

I think this is the reason:

The antiderivative is:

So to do the definite integral you would end up subtracting 2 infinities. That is why the integral is undefined.

Last edited by bobbym (2009-11-22 22:29:11)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gckc123

Member

Registered: 2009-10-19

Posts: 15

Re: expected value of a pdf

the distribution you stated is called Cauchy distribution.
Wikipedia has a very good explanation for why the mean is undefined.

http://en.wikipedia.org/wiki/Cauchy_distribution

Read the section "Explanation of undefined moments"

Cheers

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Maths is fun!

gckc123

Member

Registered: 2009-10-19

Posts: 15

Re: expected value of a pdf

One of a very interesting property of Cauchy distribution is that the mode and median is 0.
But the mean is undefined.

Apparently intuition fails you here(!!) You have to look at the definition of expectation.

Cheers

Last edited by gckc123 (2009-11-22 22:32:15)

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Maths is fun!

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: expected value of a pdf

Thanks bobbym and gckc123,

I was asking because I had a similar probability density function which I thought had an expected value of infinity, but it turns out it was my calculator's fault that it was coming out as undefined!

But yeah, Cauchy distribution is quite counterintuitive... I was always taught that an even graph will have its mean at the middle.

Last edited by Identity (2009-11-23 12:30:42)