# Generating random variates inside summation

I'd like to generate a table of sums which each a non-correlated, normally distributed random variate inside. Here is the sample code:

Ir = Cos[2 Pi xr/P + 2 Pi k/n] + Subscript[\[Epsilon], k];
Io = ReplaceAll[Ir, {xr -> xo, k -> j}];
d\[Phi]measured =     FullSimplify[ArcTan[-Sum[Io Sin[2 Pi j/n], {j, 1, n}]/Sum[Io Cos[2 Pi j/n], {j, 1, n}]] - ArcTan[-Sum[Ir Sin[2 Pi k/n], {k, 1, n}]/Sum[Ir Cos[2 Pi k/n], {k, 1, n}]]];
nval = 4;
numvars = 4;
a = Table[
ReplaceAll[d\[Phi]measured, { xo -> 0.1, xr -> 0, P -> 10, n -> nval}],
{x, 1,numvars}]


When this returns, I want each Epsilon to be a random variate, but I'm not sure how to generate a new random variate for each of the values.

I found a slight workaround by adding: /. {x_Subscript :> RandomVariate[NormalDistribution[0, .001]]} after the output of the summation, I was able to get the random variables to replace all of the subscripted epsilons

• However, $\epsilon_1$ in the first term will not be the same as $\epsilon_1$ in the second term. They will independent random numbers. – C. E. Aug 15 '19 at 22:15
• This is actaully what I was looking for, but my wording did indeed convey that I wanted each subscripted term to match. Thanks for you solution, it may help others. – Niall O'Dowd Aug 17 '19 at 23:06

This is a simple solution, similar to your own except that with this solution all $$\epsilon$$ with the same index will be replaced by the same random number.

values = Table[
Subscript[\[Epsilon], k] -> RandomVariate[NormalDistribution[0, 0.001]],
{k, numvars}
];
a /. values


{0.0629043, 0.0629043, 0.0629043, 0.0629043}

• Thanks for the response! My application requires each and every epsilon to be a different variate, but this solution may be useful if I need to match the subscripts – Niall O'Dowd Aug 16 '19 at 23:07