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

𝒲=WeightedData[data,wts];
{Mean[𝒲],Variance[𝒲]}

𝒟=EstimatedDistribution[𝒲,NormalDistribution[μ,σ]];
{Mean[𝒟],Variance[𝒟]}

𝒟NP=HistogramDistribution[𝒲];
{Mean[𝒟],Variance[𝒟]}

Labeled[Plot[Evaluate@{PDF[𝒟,t],PDF[𝒟NP,t]},{t,-4,4},Exclusions->None,Filling->Axis,Axes->{True,False},ImageSize->450,PlotStyle->Thick,PlotLegends->Placed[Style[#,Bold,FontFamily->"Helvetica"]&/@{"NormalDistribution","HistogramDistribution"},Bottom],Frame->True,GridLines->Automatic,GridLinesStyle->Directive[Gray,Dotted]],Column[{Style["Parametric and Nonparametric Estimates",Bold,FontFamily->"Helvetica",FontSize->18],Style["Weighted Data",Bold,FontFamily->"Helvetica",FontColor->Gray,FontSize->14]}],{{Top,Left}}]