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 trials Each trial has p chance of winning |
p,n | n p | n (1-p) p | |

Geometric | Number of trials needed to to obtain a success, Each trial has p chance of success |
p | |||

Negative Binomial | Number of trials needed to to obtain r successes, Each trial has p chance of success |
r,p | |||

Hypergeometric | Number of black balls drawn from urn when taking m balls without replacement. urn has total of n balls r 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 trials Each trial has p chance of winning |
p,n | n p | n (1-p) p | |

Geometric | Number of trials needed to to obtain a success, Each trial has p chance of success |
p | |||

Negative Binomial | Number of trials needed to to obtain r successes, Each trial has p chance of success |
r,p | |||

Hypergeometric | Number of black balls drawn from urn when taking m balls without replacement. urn has total of n balls r 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) |