TransformedDistribution, distribution of a mean

• To: mathgroup at smc.vnet.net
• Subject: [mg120710] TransformedDistribution, distribution of a mean
• From: paulvonhippel at yahoo <paulvonhippel at yahoo.com>
• Date: Sat, 6 Aug 2011 02:12:44 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I'm getting another odd result using TransformedDistribution.

I define a variable as the mean of M iid standard normal variables.
When I submit the problem in the following way, Mathematica tells me
that the mean is also distributed standard normal.

In[1]=    meanZ = TransformedDistribution[Sum[Subscript[x, i]/M, {i,
1, M}],
Subscript[x, i] \[Distributed] NormalDistribution[0, 1]]

Out[1]= NormalDistribution[0, 1]

This is wrong, of course. The mean of M iid standard normal variables
is distributed NormalDistribution[0,1/Sqrt[M]].

Why is Mathematica not giving me that answer?

Many thanks,
Paul

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