Uniform Product Distribution

Author

Eric W. Weisstein
January 10, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plots

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

a=Table[Random[],{10^4}];
b=Table[Random[],{10^4}];

BarChart[BinCounts[a,{0,1,.05}],PlotRange->All,BarSpacing->0]

BarChart[BinCounts[b,{0,1,.05}],PlotRange->All,BarSpacing->0]

BarChart[BinCounts[a b,{0,1,.05}],PlotRange->All,BarSpacing->0]

Integrals

2

Integrate[DiracDelta[x y-z],{x,0,1},{y,0,1}]

Integrate[DiracDelta[x y-z],{x,0,1},{y,0,1},GenerateConditions->False]

V7

Integrate[DiracDelta[x y-z],{x,0,1},{y,0,1},Assumptions->z>0]

3

V5.0

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0,GenerateConditions->False]

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0]

V5.1

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0,GenerateConditions->False]

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0]

V5.2

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0,GenerateConditions->False]

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0]

V6

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0,GenerateConditions->False]//Timing

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0]//Timing

V7

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0,GenerateConditions->False]//Timing

Integrate[DiracDelta[x y z-t],{x,0,1},{y,0,1},{z,0,1},Assumptions->x>0&&t>0]//Timing

4

V5.0

Integrate[DiracDelta[x y z w-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},GenerateConditions->False,Assumptions->x>0&&t>0]

V7

Integrate[DiracDelta[x y z w-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},GenerateConditions->False,Assumptions->x>0&&t>0]//Timing

Integrate[DiracDelta[x y z w-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},Assumptions->x>0&&t>0]//Timing

5

Integrate[DiracDelta[x y z w v-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},{v,0,1},GenerateConditions->False]

v5

Integrate[DiracDelta[x y z w v-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},{v,0,1},GenerateConditions->False,Assumptions->x>0&&t>0]

v5.0.1

Integrate[DiracDelta[x y z w v-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},{v,0,1},GenerateConditions->False,Assumptions->x>0&&t>0]

V7

Integrate[DiracDelta[x y z w v-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},{v,0,1},GenerateConditions->False,Assumptions->x>0&&t>0]//Timing

Integrate[DiracDelta[x y z w v-t],{x,0,1},{y,0,1},{z,0,1},{w,0,1},{v,0,1},Assumptions->x>0&&t>0]//Timing

Product of n

Integrate[Log[t]^n,{t,0,1},Assumptions->n>0]

Table[Log[x]^n/n!,{n,4}]

Integrate[(-1)^n Log[x]^n/n!,{x,0,1},Assumptions->n>0&&n∈Integers]

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

Plot[Evaluate[Table[(-1)^(n-1) Log[u]^(n-1)/(n-1)!,{n,2,6}]],{u,0,1},
PlotStyle->{Red,Yellow,Green,Blue,Violet},
PlotRange->{0,3},
AxesLabel->TraditionalForm/@{u,ylabel}]