Conditional expectation

**akbar** · January 6th 2010, 01:40 PM

Let $\xi_1,\xi_2,\xi_3$ be three independent, identically distributed and bounded random variables with density $p(x)$ .

What is the distribution of: $\mathbb{E}\big(\max(\xi_1,\xi_2,\xi_3)|\min(\xi_1, \xi_2,\xi_3)\big)$ , as a function of $p$ ?

One can easily obtain the distribution of $\max(\xi_1,\xi_2,\xi_3)$ and $\min(\xi_1,\xi_2,\xi_3)$ . But how do you get the joint distribution?
Unless there is a better way to proceed.

Thanks in advance for your help.

**Laurent** · January 6th 2010, 02:12 PM

Let $M=\max(\xi_1,\xi_2,\xi_3)$ and $m=\min(\xi_1,\xi_2,\xi_3)$ . You can start from there:

$P(t\leq m\mbox{ and }M\leq u)=P(t\leq \xi_1\leq u)^3$ .

This gives the distribution of $(m,M)$ .

**matheagle** · January 6th 2010, 03:15 PM

you can use the 'multinomial' formula for joint densities of order stats

$f(x_1,x_3)=3!f(x_1)[F(x_3)-F(x_1)]f(x_3)$

where I'm using $x_1$ as the first order stat and $x_3$ as the third.

**akbar** · January 6th 2010, 03:27 PM

This is what you get using the joint probability suggested by Laurent.

I am not familiar with the multinomial formula you mention. Is there any reference you could suggest where I can find it?

Thanks for your answer.

**matheagle** · January 6th 2010, 03:34 PM

It's in many books, Hogg and Craig for one.
I googled it the other day and I only found it online at another forum.
It was incorrect initially but someone finally fixed it.
sadly wikipedia only has this for uniforms...
http://en.wikipedia.org/wiki/Order_statistic
but from that you can see the full formula.
You can obtain the joint density of any subset of order stats directly.

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