Normal Sum Distribution

Author

Eric W. Weisstein
November 3, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/NormalSumDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/NormalSumDistribution.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Derivation

Integrate[((Exp[-(x-μx)^2/(2σx^2)])/(σx Sqrt[2π]))((Exp[-(y-μy)^2/(2σy^2)])/(σy Sqrt[2π]))DiracDelta[x+y-z],{x,-∞,∞},{y,-∞,∞},Assumptions->σx>0&&σy>0]

((E^(-((((-z+μx+μy))^2)/(2 (((σx)^2)+((σy)^2))))))/(FullSimplify[(Sqrt[2 π]) σx (Sqrt[(1/((σx)^2))+(1/((σy)^2))]) σy,σx>0&&σy>0]))

Plot

<<Statistics`DataManipulation`
<<Graphics`Graphics`

data=Sqrt[-2Log[#1]]Through[{Cos,Sin}[2π#2]]&@@@Table[Random[],{10^5},{2}];

BarChart[BinCounts[Plus@@@data,{-4,4,.1}],PlotRange->All,BarSpacing->0]