Issuu on Google+

Negative Binomial (n,p) P(X=x|k,p) (

)

EX=

(

)

(

, VarX=

) X=0,1,2,..0 p 1 (

Hypergeometric (M,N,K) P(X=x|N,M,K)=

Discrete Uniform P(X=x|N)= EX=

)

, VarX=

(

)(

)

( )( ( )

λ=n(1-p) n +

) )

P=M/N,n=Kλ +

a+b +

EX= ,VarX=

)( (

Beta Binomial

n=1

, EX=

(

VarX=

a=b=1

Geometric (p) P(X=x|k,p) ( )

Σxi

minXi

λ=np n +

Poisson (λ) Pareto (a,b),EX= f(x|α,β)=

,b.1

,VarX=(

P(X=x|λ)= ) ( (

Logistic f(x|μ,β)=

) (

)

)

,

LogarithmNormal (μ,σ2) f(χ|μ,σ2)=

λ=σ2 λ +

,x

μ=νp σ =np(1-p) + 2 Ν(μ,σ ) Beta(α,β)

)

a=b +

√ EX=μ, VarX=σ2

ΕΧ= )

2

2

μ=ρλ,σ =rλ . +

μ+σΧ

ΣXi

f(x,|α,β)

( ) ( ) (

(

)

, VarX=(

Β(α,β)=

n=1

Bernoulli (p),p(x=X|P) = ( ) ,x=0,1 Ex=p , VarX=p(1-P)

(

, (

)

2

(

logx

(

Ex=np , VarX=np(1P)

Normal Ν(μ,σ2)

)

EX=

( )

Σxi

(

Binomial (n.p) P(X=x|n,p)=

ΕX=λ , VarX=λ

)

ΕΧ=μ,VarX=

VarX=

)

)

)

) (

)

(

)

Normal Ν(0,1) f(x)=

Gamma (a,b)=f(x|a,b)= √

a=b=1

, x 0,a>0,b>0

( )

Uniform f(x|a,b)=

EX=ab , VarX=ab2,X2 αν a=1

EX=

Σxi2 Cauchy (θ,σ) f(x|θ,σ)==

( )

)

δεν υπάρχει ΣXi

t (ν) f(x|ν)= (

(

)

F-Fisher ,x>0

( ) (

(

) X2

,EX=b,VarX=b2

minXi

)

(

γ=1

)

Weibull (γ,λ ) ,x 0, β>0,γ>0 f(x|γ,β)=

-λlogx

Exponentia(λ),λ= ,x f(x,b)=

νiΧ +

( ) ( ) )

( )

λ=2 , ν=2

F(ν1,ν2)=f(x|ν1,ν2)=

t - Student

)

,x

( )

+ n=1

Σxi

a=1

EX=ν , VarX=2ν

ΕΧ, VarX , ρ.γ.σ

(

a= , b=2

Xi Square(v), f(x|ν)= (

,VarX=

X1-X2 |X| | Double Exponential F(x,μ,σ2)= σ>0,ΕΧ=μ,VarX=2σ2

Limit Transformation or Special case

Fitting from Leemis 1986


2relationship all distributions cont work