Scott Dale Gilbert wrote:
>
> Let x be uniformly distributed on an interval (say [-1,1]). Let
> y = w1 x1 + w2 x2 + ... + wn xn be a weighted average of
> random variables x1,x2,...,xn which themselves are independent and
> identically
> distributed draws from the distribution of x. Let the weights wi, i =
> 1,...,n,
> sum to 1, and satisfy the bounds 0 < wi < 1.
>
> Conjecture: The random variable y has a probability density function
> which is a polynomial spline function of order n-1.
>
> Is this conjecture true, and if so, are there known closed-form (in
> terms of w, n) solutions
> for the density of y?
Yes, it's true. Easy proof: f_Y(y) is proportional to the area of the
intersection of the w1 x w2 x ...x wn hyperparallellopiped (*love* that
word!) with the plane x1 + x2 + ... + xn = y; and this
is an interior-disjoint union of n-simplexes.
When the weights differ, the function f_Y(y) can be very complicated
(typically with 2^n+1 spline sections including the identically-zero
regions above & below). However,it is (n-2) times differentiable at
each node (including the endpoints) (this follows from
inclusion-exclusion and the fact that the first (n-2) derivatives of
y^(n-1) are 0 when y=0).
-Robert Dawson
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
