Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

706 Common Distributions

Continuous Distributions, Continued

Laplace (2.2.4)

−∞<θ<∞ f(x)=^12 e−|x−θ|, −∞<x<∞

μ=θ, σ^2 =2

m(t)=etθ 1 −^1 t 2 , − 1 <t< 1

Logistic (6.1.8)

−∞<θ<∞ f(x)=(1+expexp{−{−((xx−−θθ)})}) 2 , −∞<x<∞

μ=θ, σ^2 =π

2

3

m(t)=etθΓ(1−t)Γ(1 +t), − 1 <t< 1

Normal,N(μ, σ^2 ) (3.4.6)

−∞<μ<∞ f(x)=√ 21 πσexp

{

−^12

(x−μ

σ

) 2 }

, −∞<x<∞

σ> 0

μ=μ, σ^2 =σ^2

m(t)=exp{μt+(1/2)σ^2 t^2 }, −∞<t<∞

t,t(r) (3.6.2)

r> 0 f(x)=Γ[(√πrr+1)Γ(r//2)2](1+x (^2) /r^1 )(r+1)/ 2 , −∞<x<∞
Ifr>1,μ=0. Ifr>2,σ^2 =r−r 2.
The mgf does not exist.
The parameterris called the degrees of freedom.
Uniform (1.7.4)
−∞<a<b<∞ f(x)=b−^1 a, a<x<b
μ=a+ 2 b,σ^2 =(b−a)
2
12
m(t)=e
bt−eat
(b−a)t, −∞<t<∞