The sum of Two Identical Normals is Normal: Difference between revisions

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== More Generally ==
== More Generally ==


More generally, [http://mathworld.wolfram.com/NormalSumDistribution.html the sum of two normals is normal], with parameters
More generally, [http://mathworld.wolfram.com/NormalSumDistribution.html the sum of two normals is normal], with parameters mean


<center><math>\left.\mu_{X+Y}=\mu_{X}+\mu_{Y}\right.</math></center>
<center><math>\left.\mu_{X+Y}=\mu_{X}+\mu_{Y}\right.</math></center>
and
and variance
<center><math>\sigma_{X+Y}^2=\sigma_{X}^2+\sigma_{Y}^2</math></center>
<center><math>\sigma_{X+Y}^2=\sigma_{X}^2+\sigma_{Y}^2</math></center>



Latest revision as of 01:01, 28 February 2009

Normal Density function

Normally distributed variable x with mean and standard deviation has density function

The Distribution of the Sum of Two iid Normal Variables

Consider the sum s of two of these random variables x. The density of s is given by the convolution of the densities of the two:

 

 

 

 

 

 

 

Conclusion: the sum is normally distributed, with mean , and with standard deviation .


More Generally

More generally, the sum of two normals is normal, with parameters mean

and variance

By induction, the sum of n normals will be normal, with parameters

and