Table of distributions properties

by Nasser Abbasi, generated using Mathematica 6.0 .1

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

 Name X= pmf P(X=K) params E(X) Var(X) Bernulli Number of wins on this trial p p (1-p) p Binomial Number of wins in n trialsEach trial has p chance of winning p,n n p n (1-p) p Geometric Number of trials needed toto obtain a success, Each trial has p chance of success p Negative Binomial Number of trials needed toto obtain r successes, Each trial has p chance of success r,p Hypergeometric Number of black balls drawnfrom urn when taking m balls withoutreplacement. urn has total of n ballsr black and m white m,r,n Poisson Number of events in given period λ λ λ

 Name X= pmf P(X=K) params E(X) Var(X) Bernulli Number of wins on this trial p p (1-p) p Binomial Number of wins in n trialsEach trial has p chance of winning p,n n p n (1-p) p Geometric Number of trials needed toto obtain a success, Each trial has p chance of success p Negative Binomial Number of trials needed toto obtain r successes, Each trial has p chance of success r,p Hypergeometric Number of black balls drawnfrom urn when taking m balls withoutreplacement. urn has total of n ballsr black and m white m,r,n Poisson Number of events in given period λ λ λ

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

 Name X= pdf f(x) params E(X) Var(X) Normal μ,σ μ Exponential λ Gamma α,β α β ChiSquare n n 2 n Chi n Uniform min,max Cauchy a,b Indeterminate Indeterminate Beta α,β ExtremeValue α,β α+γ β Gumbel α,β α-γ β Laplace μ,β μ HalfNormal θ

Table of expected value of  functions of random variable

 Name Y, Function of random variable X E(Y) X=Normal X μ X=Normal 2 X 2 μ X=Normal X=Normal X=Normal X=Poisson 2 X 2 λ X=Poisson X=Poisson X=Poisson X=Poisson X=Poisson X=Poisson λ X=Gamma(α,β) 2 X 2 α β X=Gamma(α,β) X=Gamma(α,β) X=Gamma(α,β) X=Gamma(α,β) X=Gamma(α,β) X=Gamma(α,β) α β X=ChiSquare(n) X n n (n+2) X=ChiSquare(1) X 1 3 2 X=ChiSquare(1) 2 X 2 12 8 X=ChiSquare(2) X 2 8 4 X=ChiSquare(2) 2 X 4 32 16 X=T(n) X X=StudentTDistribution(1) X ExpectedValue[x,StudentTDistribution[1],x] X=StudentTDistribution(1) 2 X ExpectedValue[2 x,StudentTDistribution[1],x] X=StudentTDistribution(2) X 0 X=StudentTDistribution(2) 2 X 0

Some formulas

 Var(X)=Cov(X,X) 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) E(X+Y)=E(X)+E(Y) M'(t=0)=E(X) Theorem B, page 138: Var(Y)=Var(E(Y|X))+E(Var(Y|X))

Table of moment generating functions

 Distribution M'(t=0)=E(x) Binomial n p Geometric NegativeBinomial Hypergeometric Poisson λ Normal μ Exponential Gamma α β ChiSquare n Chi -n Uniform Cauchy a-b Beta Hypergeometric1F1[α,α+β,t] ExtremeValue α+EulerGamma β Gumbel α-EulerGamma β Laplace μ HalfNormal

 Created by Wolfram Mathematica 6.0 for Students - Personal Use Only  (02 February 2008)