Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

704 Common Distributions

List of Common Discrete Distributions

Bernoulli (3.1.1)

0 <p< 1 p(x)=px(1−p)^1 −x,x=0, 1

μ=p, σ^2 =p(1−p)

m(t)=[(1−p)+pet], −∞<t<∞

Binomial (3.1.2)

0 <p< 1 p(x)=

(n

x

)

px(1−p)n−x,x=0, 1 , 2 ,...,n

n=1, 2 ,...

μ=np, σ^2 =np(1−p)

m(t)=[(1−p)+pet]n, −∞<t<∞

Geometric (3.1.4)

0 <p< 1 p(x)=p(1−p)x,x=0, 1 , 2 ,...

μ=pq,σ^2 =^1 −p 2 p

m(t)=p[1−(1−p)et]−^1 ,t<−log(1−p)

Hypergeometric(N, D, n) (3.1.7)

n=1, 2 ,...,min{N, D} p(x)=

(Nn−−Dx)(Dx)

(Nn)

,x=0, 1 , 2 ,...,n

μ=nND,σ^2 =nDNNN−DNN−−n 1

The above pmf is the probability of obtainingxDs

in a sample of sizen, without replacement.

Negative Binomial (3.1.3)

0 <p< 1 p(x)=

(x+r− 1

r− 1

)

pr(1−p)x,x=0, 1 , 2 ,...

r=1, 2 ,...

μ=rqp,σ^2 =r(1p− 2 p)

m(t)=pr[1−(1−p)et]−r,t<−log(1−p)

Poisson (3.2.1)

m> 0 p(x)=e−mm

x

x!,x=0,^1 ,^2 ,...

μ=m, σ^2 =m

m(t)=exp{m(et−1)}, −∞<t<∞