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TransformedDistribution, for a sum of M iid variables

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120414] TransformedDistribution, for a sum of M iid variables
  • From: paulvonhippel at yahoo <paulvonhippel at yahoo.com>
  • Date: Thu, 21 Jul 2011 21:07:26 -0400 (EDT)

Using the TransformedDistribution function it is easy to show that the
sum of two normal variables is normal, e.g.,

Input:
TransformedDistribution[Z1+Z2,{Z1\
[Distributed]NormalDistribution[0,1],Z2\
[Distributed]NormalDistribution[0,1]}]

Output:
NormalDistribution[0, Sqrt[2]]


Likewise it wouldn't be hard to show that the sum of 3 normal
variables is normal, or 4.

But how do I show that the sum of M normal variables is normal, where
M is an arbitrary positive integer.

I'm thinking I should use the Sum function inside
TransformedDistribution, but I don't know how the notation would work.
I see that there's a UniformSumDistribution function, but that's
limited to uniform variables.

Later I will want to do similar calculations for nonnormal
distributions.

Thanks for any pointers.


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