Table of distributions properties

by Nasser Abbasi, generated using Mathematica 6.0 .1

Table of discrete distributions functions, E (X), Var (X)

Grid[{ {"Name","X=","pmf P(X=K)","params","E(X)","Var(X)"},

{Text["Bernulli"],Text["Number of wins on this trial"],PDF[BernoulliDistribution[p],k],"p",Mean[BernoulliDistribution[p]],Variance[BernoulliDistribution[p]]},

{Text["Binomial"],Text["Number of wins in n trials\nEach trial has p chance of winning"],PDF[BinomialDistribution[n,p],k],"p,n",Mean[BinomialDistribution[n,p]],Variance[BinomialDistribution[n,p]]},

{Text["Geometric"],Text["Number of trials needed to\nto obtain a success, Each trial has\n p chance of success"],PDF[GeometricDistribution[p],k],"p",Mean[GeometricDistribution[p]],Variance[GeometricDistribution[p]]},

{Text["Negative Binomial"],Text["Number of trials needed to\nto obtain r successes, Each trial has\n p chance of success"],PDF[NegativeBinomialDistribution[r,p],k],"r,p",Mean[NegativeBinomialDistribution[r,p]],Variance[NegativeBinomialDistribution[r,p]]},


{Text["Hypergeometric"],Text["Number of black balls drawn\nfrom urn when taking m balls without\nreplacement. urn has total of n balls\nr black and m white"],PDF[HypergeometricDistribution[m,r,n],k],"m,r,n",Mean[HypergeometricDistribution[m,r,n]],Variance[HypergeometricDistribution[m,r,n]]},

{Text["Poisson"],Text["Number of events in given period"],PDF[PoissonDistribution[λ],k],"λ",Mean[PoissonDistribution[λ]],Variance[PoissonDistribution[λ]]}

},
Frame->All,Alignment->Left]//TraditionalForm

Table of continuous distributions functions, E (X), Var (X)

Grid[{ {"Name","X=","pdf f(x)","params","E(X)","Var(X)"},

{Text["Normal"],Text[" "],PDF[NormalDistribution[μ,σ],x],"μ,σ",Mean[NormalDistribution[μ,σ]],Variance[NormalDistribution[μ,σ]]},

{Text["Exponential"],Text[" "],PDF[ExponentialDistribution[λ],x],"λ",Mean[ExponentialDistribution[λ]],Variance[ExponentialDistribution[λ]]},

{Text["Gamma"],Text[" "],PDF[GammaDistribution[α,β],x],"α,β",Mean[GammaDistribution[α,β]],Variance[GammaDistribution[α,β]]},

{Text["ChiSquare"],Text[" "],PDF[ChiSquareDistribution[n],x],"n",Mean[ChiSquareDistribution[n]],Variance[ChiSquareDistribution[n]]},

{Text["Chi"],Text[" "],PDF[ChiDistribution[n],x],"n",Mean[ChiDistribution[n]],Variance[ChiDistribution[n]]},

{Text["Uniform"],Text[" "],PDF[UniformDistribution[{min,max}],x],"min,max",Mean[UniformDistribution[{min,max}]],Variance[UniformDistribution[{min,max}]]},

{Text["Cauchy"],Text[" "],PDF[CauchyDistribution[a,b],x],"a,b",Mean[CauchyDistribution[a,b]],Variance[CauchyDistribution[a,b]]},

{Text["Beta"],Text[" "],PDF[BetaDistribution[α,β],x],"α,β",Mean[BetaDistribution[α,β]],Variance[BetaDistribution[α,β]]},


{Text["ExtremeValue"],Text[" "],PDF[ExtremeValueDistribution[α,β],x],"α,β",Mean[ExtremeValueDistribution[α,β]],Variance[ExtremeValueDistribution[α,β]]},

{Text["Gumbel"],Text[" "],PDF[GumbelDistribution[α,β],x],"α,β",Mean[GumbelDistribution[α,β]],Variance[GumbelDistribution[α,β]]},

{Text["Laplace"],Text[" "],PDF[LaplaceDistribution[μ,β],x],"μ,β",Mean[LaplaceDistribution[μ,β]],Variance[LaplaceDistribution[μ,β]]},

{Text["HalfNormal"],Text[" "],PDF[HalfNormalDistribution[θ],x],"θ",Mean[HalfNormalDistribution[θ]],Variance[HalfNormalDistribution[θ]]}


},
Frame->All,Alignment->Left]//TraditionalForm

Table of expected value of  functions of random variable

Grid[{ {"Name","Y, Function of random variable X","E(Y)","E[(Y^2)]","Var[Y]=E((Y^2))-[E(Y)]^2"},

{Text["X=Normal"],Text["X"],(z=ExpectedValue[x,NormalDistribution[μ,σ],x]),z2=ExpectedValue[(((x))^2),NormalDistribution[μ,σ],x],z2-z^2},
{Text["X=Normal"],Text["2 X"],(z=ExpectedValue[2 x,NormalDistribution[μ,σ],x]),z2=ExpectedValue[(((2 x))^2),NormalDistribution[μ,σ],x],z2-z^2},
{Text["X=Normal"],Text["(X^2)"],(z=ExpectedValue[(x^2),NormalDistribution[μ,σ],x]),z2=ExpectedValue[((((x^2)))^2),NormalDistribution[μ,σ],x],z2-z^2},
{Text["X=Normal"],Text["(X^3)"],(z=ExpectedValue[(x^3),NormalDistribution[μ,σ],x]),z2=ExpectedValue[((((x^3)))^2),NormalDistribution[μ,σ],x],z2-z^2},
{Text["X=Normal"],Text["(X^4)"],(z=ExpectedValue[(x^4),NormalDistribution[μ,σ],x]),z2=ExpectedValue[((((x^4)))^2),NormalDistribution[μ,σ],x],z2-z^2},


{Text["X=Poisson"],Text["2 X"],(z=ExpectedValue[2 x,PoissonDistribution[λ],x]),z2=ExpectedValue[(((2 x))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["(X^2)"],(z=ExpectedValue[(x^2),PoissonDistribution[λ],x]),z2=ExpectedValue[((((x^2)))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["(X^3)"],ExpectedValue[(x^3),PoissonDistribution[λ],x],z2=ExpectedValue[((((x^3)))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["(X^4)"],(z=ExpectedValue[(x^4),PoissonDistribution[λ],x]),z2=ExpectedValue[((((x^4)))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["(1/(1 + X))"],(z=ExpectedValue[(1/(1+x)),PoissonDistribution[λ],x]),z2=ExpectedValue[((((1/(1+x))))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["\!\(\*FractionBox[\(1\), \(1 + 2\\ X\)]\)"],(z=ExpectedValue[(1/(1+2x)),PoissonDistribution[λ],x]),z2=ExpectedValue[((((1/(1+2x))))^2),PoissonDistribution[λ],x],z2-z^2},
{Text["X=Poisson"],Text["(Sqrt[X])"],(z=ExpectedValue[ (Sqrt[x]),PoissonDistribution[λ],x]),z2=ExpectedValue[((((Sqrt[x])))^2),PoissonDistribution[λ],x],z2-z^2},

{Text["X=Gamma(α,β)"],Text["2 X"],(z=ExpectedValue[2 x,GammaDistribution[α,β],x]),z2=ExpectedValue[(((2 x))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["(X^2)"],(z=ExpectedValue[(x^2),GammaDistribution[α,β],x]),z2=ExpectedValue[((((x^2)))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["(X^3)"],(z=ExpectedValue[(x^3),GammaDistribution[α,β],x]),z2=ExpectedValue[((((x^3)))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["(X^4)"],(z=ExpectedValue[(x^4),GammaDistribution[α,β],x]),z2=ExpectedValue[((((x^4)))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["(1/(1 + X))"],(z=ExpectedValue[(1/(1+x)),GammaDistribution[α,β],x]),z2=ExpectedValue[((((1/(1+X))))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["\!\(\*FractionBox[\(1\), \(1 + 2\\ X\)]\)"],(z=ExpectedValue[(1/(1+2x)),GammaDistribution[α,β],x]),z2=ExpectedValue[((((1/(1+2 X))))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=Gamma(α,β)"],Text["(Sqrt[X])"],(z=ExpectedValue[ (Sqrt[x]),GammaDistribution[α,β],x]),z2=ExpectedValue[((((Sqrt[x])))^2),GammaDistribution[α,β],x],z2-z^2},
{Text["X=ChiSquare(n)"],Text["X"],(z=ExpectedValue[ x,ChiSquareDistribution[n],x]),z2=ExpectedValue[(((x))^2),ChiSquareDistribution[n],x],z2-z^2},
{Text["X=ChiSquare(1)"],Text["X"],(z=ExpectedValue[ x,ChiSquareDistribution[1],x]),z2=ExpectedValue[(((x))^2),ChiSquareDistribution[1],x],z2-z^2},
{Text["X=ChiSquare(1)"],Text["2 X"],(z=ExpectedValue[2  x,ChiSquareDistribution[1],x]),z2=ExpectedValue[(((2 x))^2),ChiSquareDistribution[1],x],z2-z^2},
{Text["X=ChiSquare(2)"],Text["X"],(z=ExpectedValue[ x,ChiSquareDistribution[2],x]),z2=ExpectedValue[(((x))^2),ChiSquareDistribution[2],x],z2-z^2},
{Text["X=ChiSquare(2)"],Text["2 X"],(z=ExpectedValue[2  x,ChiSquareDistribution[2],x]),z2=ExpectedValue[(((2 x))^2),ChiSquareDistribution[2],x],z2-z^2},
{Text["X=T(n)"],Text["X"],(z=ExpectedValue[x,StudentTDistribution[n],x]),z2=ExpectedValue[(((2 x))^2),StudentTDistribution[n],x],z2-z^2},
{Text["X=StudentTDistribution(1)"],Text["X"],(z=ExpectedValue[ x,StudentTDistribution[1],x]),z2=ExpectedValue[(((x))^2),StudentTDistribution[1],x],z2-z^2},
{Text["X=StudentTDistribution(1)"],Text["2 X"],(z=ExpectedValue[2  x,StudentTDistribution[1],x]),z2=ExpectedValue[(((2 x))^2),StudentTDistribution[1],x],z2-z^2},
{Text["X=StudentTDistribution(2)"],Text["X"],(z=ExpectedValue[ x,StudentTDistribution[2],x]),z2=ExpectedValue[(((x))^2),StudentTDistribution[2],x],z2-z^2},
{Text["X=StudentTDistribution(2)"],Text["2 X"],(z=ExpectedValue[2  x,StudentTDistribution[2],x]),z2=ExpectedValue[(((2 x))^2),StudentTDistribution[2],x],z2-z^2}

},
Frame->All,Alignment->Left]//TraditionalForm

Some formulas

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{"Var(X)=Cov(X,X)","Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)",""},
{"Cov(X,Y)=E[(X-(Subscript[μ, x]))(Y-(Subscript[μ, y]))]","Cov(X,Y)=E(XY)-E(X)E(Y)","Cov(a+X,Y)=Cov(X,Y)"},
{"Cov(aX,bY)=ab Cov(X,Y)","Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)","ρ(X,Y)=((Cov (X, Y))/(Sqrt[Var (X) Var (Y)]))"},
{"E(X)=∫x f(x)\[DifferentialD]x","E(X)=(∑_allx^n)x p(x)","E(X+Y)=E(X)+E(Y)"},
{"moment generating: (Subscript[M, X])(t)=E((E^xt))=(∫_(-∞)^∞)(E^xt)(Subscript[f, X])(x)\[DifferentialD]x","M'(t=0)=E(X)","M''(t=0)=E((X^2))"},
{"Theorem A, page 137: E(Y) = E(E(Y|X))","Theorem B, page 138: Var(Y)=Var(E(Y|X))+E(Var(Y|X))","E(X|Y)=∫x (Subscript[f, X | Y])(x|y) \[DifferentialD]x"},
{"conditional Var: Var(Y|X) = E((Y^2)|X)-(E(Y|X))^2","Chebyshev inequality: P(|X-μ|>t)≤((σ^2)/(t^2))","(e^x)=(∑_(n = 0)^∞)((x^n)/(n!))=1+x+((x^2)/(2!))+..."},
{"(lim_(n -> ∞))(1+(x/n))^nb=(e^xb)","(lim_(n -> ∞))(1+(x/n))^n=(e^x)","ln(x)=(∫_1^x)(1/u)\[DifferentialD]u"},{"ln(e)=(∫_1^e)(1/u)\[DifferentialD]u=1","Law of total expectation: E(Y)=(∑_(i = 1)^n)E(Y|X=(Subscript[x, i]))P(X=(Subscript[x, i]))","Random Sums, where N is random: T=(∑_(i = 1)^N)(Subscript[X, i]) \nE(T)=E(N)E(X) and Var(T)=[E(X)]^2Var(N)+E(N)Var(X)"},
{"((N (0, 1))/(Sqrt[((ChiSquare (n))/n)]))~T distribution with n degree of freedom" ,"T(1): PDF=" Text[PDF[StudentTDistribution[1],x]],"T(2): PDF=" Text[PDF[StudentTDistribution[2],x]]}
}
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Table of moment generating functions

Grid[{ {"Distribution","(Subscript[M, X])(t). moment generating function","M'(t=0)=E(x)","M''(t=0)=E((x^2))","M'''(t=0)=E((X^3))"},

{"Binomial",mgf=CharacteristicFunction[BinomialDistribution[n,p],t]/.Exp[I t]->Exp[t],Limit[D[mgf,t],t->0],(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Geometric",mgf=CharacteristicFunction[GeometricDistribution[p],t]/.Exp[I t]->Exp[t],Limit[D[mgf,t],t->0],(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"NegativeBinomial",mgf=CharacteristicFunction[NegativeBinomialDistribution[r,p],t]/.Exp[I t]->Exp[t],Limit[D[mgf,t],t->0],(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Hypergeometric",mgf=CharacteristicFunction[HypergeometricDistribution[m,r,n],t]/.Exp[I t]->Exp[t],Limit[D[mgf,t],t->0]//Simplify,"",""},

{"Poisson",mgf=CharacteristicFunction[PoissonDistribution[λ],t]/.Exp[I t]->Exp[t],Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Normal",mgf=CharacteristicFunction[NormalDistribution[μ,σ],t]/.(E^(I t μ-(((t^2) (σ^2))/2)))->(E^(t μ-(((t^2) (σ^2))/2))),Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Exponential",mgf=CharacteristicFunction[ExponentialDistribution[λ],t]/.(λ/(-I t+λ))->(λ/(- t+λ)),Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Gamma",mgf=CharacteristicFunction[GammaDistribution[α,β],t]/.(1-I t β)->(1- t β),Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"ChiSquare",mgf=CharacteristicFunction[ChiSquareDistribution[n],t]/.(1-2 I t)->(1-2 t),Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Chi",mgf=CharacteristicFunction[ChiDistribution[n],t]/.I->1,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Uniform",mgf=CharacteristicFunction[UniformDistribution[{a,b}],t]/.-((I ((E^(I b t))-(E^(I a t))))/((b-a) t))->(( -((E^(b t))-(E^( a t))))/((b-a) t)),Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Cauchy",mgf=CharacteristicFunction[CauchyDistribution[a,b],t]/.(E^(I a t-b t Sign[t]))->(E^( a t-b t Sign[t])),Assuming[Element[{a,b},Reals],Limit[D[mgf,t],t->0]]//Simplify,Assuming[Element[{a,b},Reals],(Limit[D[mgf,{t,2}],t->0])]//Expand,Assuming[Element[{a,b},Reals],(Limit[D[mgf,{t,3}],t->0])]//Expand},

{"Beta",mgf=CharacteristicFunction[BetaDistribution[α,β],t]/.I->1,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"ExtremeValue",mgf=CharacteristicFunction[ExtremeValueDistribution[α,β],t]/.(E^(I t α)) Gamma[1-I t β]->(E^(t α)) Gamma[1- t β]//Simplify,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},


{"Gumbel",mgf=CharacteristicFunction[GumbelDistribution[α,β],t]/.(E^(I t α)) Gamma[1+I t β]->(E^( t α)) Gamma[1+ t β]//Simplify,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"Laplace",mgf=CharacteristicFunction[LaplaceDistribution[μ,β],t]/.((E^(I t μ))/(1+(t^2) (β^2)))->((E^( t μ))/(1+(t^2) (β^2)))//Simplify,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand},

{"HalfNormal",mgf=CharacteristicFunction[HalfNormalDistribution[θ],t]/.(E^(-((π (t^2))/(4 (θ^2))))) (1+I Erfi[(((Sqrt[π]) t)/(2 θ))])->(E^(-((π (t^2))/(4 (θ^2))))) (1+ Erfi[(((Sqrt[π]) t)/(2 θ))])//Simplify,Limit[D[mgf,t],t->0]//Simplify,(Limit[D[mgf,{t,2}],t->0])//Expand,(Limit[D[mgf,{t,3}],t->0])//Expand}


},

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