ON THE MOMENTS OF RANDOM VARIABLES UNIFORMLY DISTRIBUTED OVER A POLYTOPE

Suppose X (X1, X2, X,) is a random vector uniformly distributed over a polytope. In this note, the author derives a formula for E(X[X...), (the expected value of X[X7...), in terms of the extreme points ofthe polytope.


INTRODUCTION
Von Hohenbalken [2] presents an algorithm for decomposing a polytope V {xlAx > b,x where A is an m x n matrix, into n-simplexes and states a formula for the center of gravity of V.In this note we explain how these results can be generalized to find formulas for E(X[X...), where X (XI,X2,..,X,.,) is a random vector (r.v.) uniformly distributed over V. Let fQ f(z)dz be the Lebesque integral of a continuous function over a compact set Q E R In order to motivate the approach we have chosen, consider the problem of finding the r th moment E(X[) .using the definition formulating the integral E(X[) Z x[dx as a set of n-fold iterated integrals and evaluating them.But how does one find the range of integration in a systematic way, given an arbitrary matrix A7 It appears that no algorithm has yet been developed.

MAIN RESULTS
In order to simplify the arguments, we assume that L(V), the Lebesque measure of V is positive In the following, we denote vectors by lower case letters with or without superscript.The th element the vector x is x, and L(Q) is the Lebesque measure of a compact set Q c R Since V is a polytope, it is the convex hull of its extreme points, say x,x , ...,x Let E be the set of these extreme points.Then there exists a set S (S, $2, Sg} of n simplexes such that the n + 1 extreme points of each S, are in E, 1, 2, g and that i) V=U,S, ii) L(V) _, L(S).
The set E can be found using the algorithm in Dyer et al [1] while the set S can be found by decomposing E using the results given in [2].Now i) and ii) imply that E E(X: X...) L(S,)E(XX...IX C S,)/L(V) a, b O, 1, Let z', j 1, 2, n + 1, be the extreme points of S,, 1, 2, 9 and let A' be the matrix whose column is z ' z'''+, j 1, 2, n, and B the matrix whose jth column is z', j 1, 2, n + 1.
Evaluating (2) for the case of E(U), it is easily seen that E(U) E(U) n!a!/(a + n)!.For the case of E(UU), it is easier to evaluate for 1 and j 2. The integrand after integrating with n!a[cl )(a+l fdu2_ (a+b/2)! from which we derive easily that E (UU b) (++n), The proof can be completed by repeating this argument An immediate consequence of the above result is that if Y (Y1, Y2, Y,+I is an v uniformly distributed over Y={ y[y>-O'i=I'2''''n+I'Ey=I then E(Y?Y2 Y,q,) n''' q! (m+,), for all il,i2,...,ir E (1,2,...,n+ 1} and for all r _< n.We now show that the result is true for r n + 1 so that E(YI'Y2...YY+I)=q , ! , ! b ! . . .q , t , ( w + ) ! : . ., where w a + b + + q + t.The proof is by induction.It is easily proved, using the substitution Y+I (1 Y1 Y2 Y) that the result is true for 1.So assume that it is true for all positive integers, a, b, p and for some t _> 2. Then simplifies, by the induction hypothesis, to the required result, proving that n!a!b!..
for all {j, k, r} C ( 1, 2, n + 1} such that j :/: k r.Now using (1) and the properties of the matrix B', it can be easily proved that E(X; Xk'...X"[X C S,) E(b'Y) (b'Y) (b'Y) " The theorem now follows immediately from (2.3).where the summation is over all non-negative integers such that p + q +... " m.

COROLLARY 1 .
Let the fh row of B be b' (b, b, b.+l ).Then E Xj X C S m + n E b jl b b.n+l