TableofDistributions
DistributionPMF/PDFandSupportExpectedValueVarianceMGF
P(X=1)=p
Bernoulli
t
Bern(p)P(X=0)=q=1pppqq+pe
n
knk
P(X=k)=pq
Binomialk
tn
Bin(n;p)k2f0;1;2;:::ngnpnpq(q+pe)
k
P(X=k)=qp
Geometric
p
2t
Geom(p)k2f0,1,2,...gq=pq=p;qe<1
t
1qe
r+n1
rn
P(X=n)=pq
NegativeBinomial
r1
p
2rt
NBin(r;p)n2f0,1,2,...grq=prq=p();qe<1
t
1qe
w+b
wb
P(X=k)==
Hypergeometricknkn
nww+bn
HGeom(w;b;n)k2f0;1;2;:::;ng=n(1)messy
b+ww+b1nn
k
e
P(X=k)=
Poissonk!
t
(e1)
Pois()k2f0,1,2,...ge
1
f(x)=
Uniformba
2
tbta
(ba)
a+bee
Unif(a;b)x2(a;b)
212t(ba)
2
2
1=(2)
(x)
p
f(x)=e
Normal2
22
t
t+
22
2
N(;)x2(1;1)e
x
f(x)=e
Exponential
11
Expo()x2(0;1);t<
2
t
1ax1
f(x)=(x)e
(a)x
Gamma
a
aa
Gamma(a;)x2(0;1);t<
2
t
(a+b)
a1b1
f(x)=x(1x)
(a)(b)
Beta
(1)
a
Beta(a;b)x2(0;1)=messy
a+b(a+b+1)
22
1
(logx)=(2)
p
e
Log-Normal
x2
22
2+=22
LN(;)x2(0;1)=e(e1)doesn’texist
1n=21x=2
xe
n=2
Chi-Square2(n=2)
2n=2
x2(0;1)n2n(12t);t<1=2
n
((n+1)=2)
2(n+1)=2
p(1+x=n)
n(n=2)
Student-t
n
tx2(1;1)0ifn>1ifn>2doesn’texist
n
n2 |
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