In article <[EMAIL PROTECTED]>,
Robert J. MacG. Dawson <[EMAIL PROTECTED]> wrote:
>John wrote:
>> Please help me solve this problem.
>> Let t and u are independent uniform distributions with range from 0 to 1,
>> and a, b, and c are constants.
>> I tried to compute the expectation of (t^a)*(u^b)*[(1-t-u)^c].
>> Since t and u are independent, finding the expectation of t and u is
>> finding the integral of (t^a)*(u^b)*[(1-t-u)^c] with respect to t and,
>> then, u.
>> I rewrote (t^a)*(u^b)*[(1-t-u)^c] to different forms but no luck.
>> Somehow, I believe the result is either Hypergeometric function or Beta
>> function.
> I know this isn't quite what you asked, but it looks to me as if it
>might be nicer integrated on 0<t, 0<u, t+u<1 - in which case I'd bet
>sight unseen on the obvious three-parameter variation of the beta
>function, Gamma(a) Gamma(b) Gamma(c) / Gamma(a+b+c).
If it is as you say, t and u are not independent, and the
resulting integral is well known as the Dirichlet integral;
however, the expression is
Gamma(a+1) Gamma(b+1) Gamma(c+1) / Gamma(a+b+c+3).
It goes over for any number of terms.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Deptartment of Statistics, Purdue University
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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