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Linear Combination of Normal Distributions
I'm just playing around with RandomVariate. Right now I can't really figure out why a linear combination with weights 0.5 of two identical normal distributions is not yielding the same distribution.
0.5*N($\mu$,$\sigma$)+0.5*N($\mu$,$\sigma$) should yield N($\mu$,$\sigma$).
This fails if I use a Monte Carlo approach.
Map[(
\[Mu] = 1;
\[Sigma] = 0.4;
p1 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
p2 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
FindDistribution[0.5*p1 + 0.5*p2]
) &, Range[10]]
I'm pretty sure Mathematica is doing everything right. Still I can't really figure out why the StandardDeviation is far away from 0.4.
Formular for the variance of the linear combination of two randomvariables:
$\operatorname{Var}( a \,X + b \, Y) = a^2 \, \operatorname{Var}(X) + b^2 \, \operatorname{Var}(Y) + 2 \, a \, b \operatorname{Cov}(X,Y)$
For the standard deviations, you have to take the squar

Formular for the variance of the linear combination of two randomvariables:
$\operatorname{Var}( a \,X + b \, Y) = a^2 \, \operatorname{Var}(X) + b^2 \, \operatorname{Var}(Y) + 2 \, a \, b \operatorname{Cov}(X,Y)$
For the standard deviations, you have to take the square root.
Your random variables are independent and have variances 0.4^2 and a = b = 0.5. So the resulting standard deviation is
Sqrt[a^2 0.4^2 + b^2 0.4^2 + 2 a b 0.]
0.282843
That's quite exactly what you obtain emperically from FindDistribution.
Lesson to learn: The distribution of the sum is not the sum of the distributions.
20180821 10:33:14 
I think the problem is the way you find the new sampled distribution.
This is what you should do for testing you expression 0.5*N(μ,σ)+0.5*N(μ,σ) ~ N(μ,σ):
Map[(\[Mu] = 1;
\[Sigma] = 0.4;
p1 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
p2 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
{Mean@#, StandardDeviation@#} &@
MixtureDistribution[{0.5, 0.5}, {FindDistribution[p1],
FindDistribution[p2]}]
) &, Range[10]]
{{1.00193, 0.39969}, {1.00019, 0.402075}, {1.00143,
0.402793}, {0.997161, 0.399242}, {1.00011, 0.400222}, {1.00281,
0.401593}, {1.00001, 0.403046}, {1.00436, 0.400939}, {0.998176,
0.402365}, {1.00206, 0.396431}}
As you can see mean values and std devs are consistent with what you would expect.
20180821 11:04:15