Particulary, I would like to find the expected value of the sum of $n$ variables distributed like ParetoDistribution[1,1.14]. For example, this works for the sum of two vars with Gamma distribution:
dist = GammaDistribution[1, 1];
c[n_] := Integrate[
InverseFourierTransform[(CharacteristicFunction[dist, t])^n, t, x]*x,
{x, -∞, ∞}]
c[2]
I've tried this and it works for almost every distribution but I think that this is not a very efficent way of doing it and still doesn't work for the Pareto.
My ultimate goal is to Plot what I've called c[n]
against n
, to see how the expected value of the sum of n
iid variables changes with n
.
I need it to do it in a Monte Carlo way, in the sense that I need to simulate each variable as a random variable but with the same underlying distribution. If I calculate the mean for a Pareto(1,1.14)
I get 8.14
but It is very uncommon that I'll get that value from a sample. So the sample mean of 2 variables is likely to be under 2*8.14
. I need to capture that randomness.