Re: TransformedDistribution, for a sum of M iid variables

• To: mathgroup at smc.vnet.net
• Subject: [mg120427] Re: TransformedDistribution, for a sum of M iid variables
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Fri, 22 Jul 2011 03:42:24 -0400 (EDT)

```On 7/21/11 at 9:07 PM, paulvonhippel at yahoo.com (paulvonhippel at
yahoo) wrote:

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

<code snipped>

>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.

Rather than approach this problem using TransformedDistribution,
I would approach this using MomentGeneratingFunction. The moment
generating function for the sum of n things selected from a
given distribution is the same as the moment generating function
for the distribution raised to the nth power. That is for the
sum of n standard normal deviates the moment generating function is:

In[3]:= MomentGeneratingFunction[NormalDistribution[], t]^n

Out[3]= (E^(t^2/2))^n

Comparing this to

In[4]:= MomentGeneratingFunction[NormalDistribution[0, s], t]

Out[4]= E^((s^2*t^2)/2)

it is clear the distribution for the sum of n deviates drawn
from a standard normal distribution is a normal distribution
with mean = 0 variance = n.

I should point out the moment generating function doesn't exist
for all distributions. In those cases you can use the
characteristic function which always exists. And since the
characteristic function is essentially the Fourier transform of
the distribution function, you can use the inverse Fourier
transform to recover the distribution function from the
characteristic function.

Use of the characteristic function or moment generating function
is a very useful technique for answering these types of
questions. Each distribution has a unique characteristic
function (and moment generating function when it exists). Also,
you can easily compute the moments of the distribution directly
from the characteristic function or moment generating function
without actually finding the distribution function.

```

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