Normal Product Distribution

Author

Eric W. Weisstein
November 3, 2003

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

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

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

Derivation

(Subscript[X, 1]), (Subscript[X, 2])

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]

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

Integrate[((BesselK[0,((Sqrt[(u^2)])/(σx σy))])/(π σx σy)),{u,-∞,∞},Assumptions->σx>0&&σy>0]

(Subscript[X, 1]), (Subscript[X, 2]), (Subscript[X, 3])

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

TraditionalForm[%]

Integrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞},Assumptions->σx>0&&σy>0&&σz>0]

With[{σx=1,σy=1,σz=1},
Integrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞}]
]

With[{σx=1,σy=1,σz=1},
Integrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞}]
]

With[{σx=1,σy=1,σz=1},
NIntegrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-10,10}]
]

(Subscript[X, 1]), (Subscript[X, 2]), (Subscript[X, 3]), (Subscript[X, 4])

With[{μx=0,μy=0,μz=0,μw=0},
Integrate[((Exp[-(x-μx)^2/(2σx^2)])/(σx Sqrt[2π]))((Exp[-(y-μy)^2/(2σy^2)])/(σy Sqrt[2π]))((Exp[-(z-μz)^2/(2σz^2)])/(σz Sqrt[2π]))((Exp[-(w-μw)^2/(2σw^2)])/(σw Sqrt[2π]))DiracDelta[x y z w-u],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},{w,-∞,∞},Assumptions->σx>0&&σy>0&&σz>0&&σw>0]
]

Plot

MakeBoxes[ordinate,TraditionalForm]:=RowBox[{SubscriptBox["P",RowBox[{SubscriptBox["X","1"],"⋯",SubscriptBox["X","n"]}]],"(","u",")"}]

Plot[{BesselK[0,Abs[u]]/Pi,
((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ))])/(2 (Sqrt[2]) (π^(3/2)) ))
},{u,-3,3},PlotStyle->{Red,Blue},
AxesLabel->TraditionalForm/@{u,ordinate}]

Integrate[((BesselK[0,((Abs[u])/(σx σy))])/(π σx σy)),{u,-∞,∞},Assumptions->σx>0&&σy>0]

Integrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞},Assumptions->σx>0&&σy>0&&σz>0]

With[{σx=1,σy=1,σz=1},
Integrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞}]
]

With[{σx=1,σy=1,σz=1},
NIntegrate[((MeijerG[{{},{}},{{0,0,0},{}},((u^2)/(8 ((σx)^2) ((σy)^2) ((σz)^2)))])/(2 (Sqrt[2]) (π^(3/2)) σx σy σz)),{u,-∞,∞}]
]

Plot

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

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

BarChart[BinCounts[Times@@@data,{-1,1,.04}],PlotRange->All,BarSpacing->0]

data1=Sqrt[-2Log[#1]]Through[{Cos,Sin}[2π#2]]&@@@Table[Random[],{10^5},{2}];

data2=Sqrt[-2Log[#1]]Through[{Cos,Sin}[2π#2]]&@@@Table[Random[],{10^5},{2}];

data=Most/@Flatten/@Transpose[{data1,data2}];

BarChart[BinCounts[Times@@@data,{-1,1,.04}],PlotRange->All,BarSpacing->0]