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3

mpsF[k_] := Mean[#] + N@Sqrt[k] (# - Mean[#]) &; (* thanks: ybeltukov *) Example: dist = BinomialDistribution[40, .1]; sample = N@RandomVariate[dist, 100]; sample2 = mpsF[10][sample]; Through@{Mean, Variance}@sample (* {4.18, 4.57333} *) Through@{Mean, Variance}@sample2 (* {4.18, 45.73333} *) Alternatively, define a function to apply a ...

2

Although this is not the solution of the problem as stated, it might be interesting to see some preliminary results, and these cannot be well read in a comment. It turns out that the problem simply is too big for a straightforward application of Mathematica. First we define some useful quantitites (we apply Mathematica nameing conventions) ...

2

Not sure what is going on with the different results of Integrate and NIntegrate. This does not mean that the analytic form of $C$ is erroneous. Note that plotting the likelihood function (using the expression of $C$ provided by Integrate and the parameter values you used) over a resticted range of $s$ (instead of $[0,1]$) clearly shows that the ...

3

The problem is numerically unstable for some parameter ranges. We shall show a simple example. Your normalized distribution is given by p[q_, n_, \[Mu]_, \[Nu]_, s_] := Exp[4 n s q] q^(4 n \[Nu] - 1) (1 - q)^(4 n \[Mu] - 1)/(Gamma[4 n \[Mu]] Gamma[ 4 n \[Nu]] Hypergeometric1F1Regularized[4 n \[Nu], 4 n (\[Mu] + \[Nu]), 4 n s]) Check ...

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