filter={1,4,6,4,1};
empirical=EmpiricalDistribution[(1/(Length[filter]))*filter->Range[1,Length[filter]]-((Length[filter]+1)/2)];
idx=Range[1,Length[filter]]-((Length[filter]+1)/2);

σ=StandardDeviation[empirical]

Show[
DiscretePlot[PDF[empirical,x],{x,idx},
PlotStyle->,
ImageSize->Large,
PlotRange->{{-5,5},{-0.02,0.42}},
AxesLabel->{"x","*"},
BaseStyle->{FontSize->14},
PlotLegends->PointLegend[{"Binomial"},LabelStyle->{FontSize->14}]
],
Plot[PDF[NormalDistribution[0,σ],x],{x,-5,5},
PlotRange->All,
PlotLegends->LineLegend[{"Gaussian"},LabelStyle->{FontSize->14}]
]
]

filter2D={filter}\[Transpose].{filter};
filter2D//MatrixForm
Total[filter2D,2]