Expectation/Integral problem

eban · science forum beginner Joined: 09 Apr 2006 Posts: 3

Hi,

Almost to embarrassed to ask, but I cannot find the solution in my calculus (binmore) or statistics (Gourieroux) books.
I want to express the expectation as integrals with cumulative distribution functions in the integrands.:
/ inf
|
EX = | x dF(x)
|
/ -inf

This should be equal to:

/ 0 / inf
| |
EX = - | F(x) dx + | (1-F(x)) dx
| |
/ -inf / 0

This should be easily proved with integration by parts, is what my book says.

I can think of:
absolute integrability:

/ 0 / inf
| |
EX = - | xf(x)dx + | xf(x)dx
| |
/ -inf / 0

integration by parts:

/ 0
| 0 |
= x.F(x) | - | F(x)dx
| -Inf |
/ -inf

/ Inf
| Inf |
+ x.F(x) | - | F(x)dx
| 0 |
/ 0

As lim(inf) F(x)=1, the second line becomes

/ inf
|
| (1-F(x)) dx
|
/ 0

which is the second part of what i am supposed to find.

In the first part I am stuck with -lim(-inf)x.F(x) which is not necessarily zero, however.
What have i done wrong ???????

Thanks for tips

Etty

eban · science forum beginner Joined: 09 Apr 2006 Posts: 3

In article <4439733c$0$11064$e4fe514c@news.xs4all.nl>,
etx@pubara.net (eban) writes:

Sorry, F is the cdf, f is the pdf, inf is infinite

A.G.McDowell · science forum beginner Joined: 17 Mar 2005 Posts: 4

In article <4439733c$0$11064$e4fe514c@news.xs4all.nl>, eban
<etx@pubara.net> writes

eban · science forum beginner Joined: 09 Apr 2006 Posts: 3

Thanks,
You solve the 'tricky' part in a neater way than I had done.
My biggest problem was the inf*0 limit. I now see it is coming from the convergence of the integral.
It feels like the case of the limit of the terms of a converging series going to zero. I shall look for a comparable theorem for integrals.

Etty

In article <ZVeBhKAB0IQEFwor@mcdowella.demon.co.uk>,
"A.G.McDowell" <mcdowella@nospam.co.uk> writes:

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