Re: Z = U(0,1)*N(0,1) + U(0,1)*N(0,1) + ... + U(0,1)*N(0,1) is distributed how?

liam_herron wrote:

If I have a r.v. Z which is the sum of n products where each product
is a Uniform r.v. multiplied by a standard normal r.v., what is the
distribution of Z? I am curious as I am simulating a factor loaded
model with random numbers.


I assume all your component random variables are independent. I doubt you will find a nice closed form expression for the distribution or density. Even with n =1, the best I can come up for the density is

z |-> integral(u=0..1, exp(-z^2 / (2 u^2)) / sqrt(2 pi)

For general n, the characteristic function of Z is

t |-> (pi / 2)^(n/2) t^-n [erf(t / sqrt(2))]^n

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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