Fit Nonparametric and Parametric Distributions to Weighted Data 

Generate random reals from to and weight them according to the PDF of a NormalDistribution using WeightedData.

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X data = RandomReal[{-5, 5}, 10^4]; wts = PDF[NormalDistribution[], data];

Create a WeightedData object and compute its mean and variance.

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X \[ScriptCapitalW] = WeightedData[data, wts]; {Mean[\[ScriptCapitalW]], Variance[\[ScriptCapitalW]]}

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Fit to a NormalDistribution using maximum likelihood estimation.

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X \[ScriptCapitalD] = EstimatedDistribution[\[ScriptCapitalW], NormalDistribution[\[Mu], \[Sigma]]]; {Mean[\[ScriptCapitalD]], Variance[\[ScriptCapitalD]]}

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Use HistogramDistribution to create a weighted histogram.

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X \[ScriptCapitalD]NP = HistogramDistribution[\[ScriptCapitalW]]; {Mean[\[ScriptCapitalD]], Variance[\[ScriptCapitalD]]}

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