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.
OK, I now think I see how to do it... Start by defining the third
variable v = 1-t-u, and integrate t^a * u^b along lines v = const.
At least for v<1 this comes out as the beta integral scaled by
(1-v), which should - I think - give something easily expressed in
B(a,b) and a power of (1-v). Then throw in the factor v^c and you
*should* have another beta integral, from which at least the triangle
integral I mentioned in ought to follow as conjectured (or similarly).
I suspect this does *not* work for the region u+t>1. Are you
*sure* you're interested in this bit? Among other things, the
restrictions on c are much stricter once you cross the line t+u=1. In
the basic triangle, we have only a,b,c > -1; beyond that line we must
have c an integer as well.
-Robert Dawson
.
.
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