Discrete univariate distributions
Distribution
Probability mass function
Range
Parameters
JE(X)
Moment-generating
Var(X)
1
Binomial 1,2
Bi(n,O)
(:)OX(I_ o)n-x
x E {O,l, ... ,n}
nEN
0<0<1
nO
nO(I- 0)
(1 - 0 + 0 exp(t))n
No. of successes in n trials
e - probability of success
Geometric
Geo(O)
OX-I(I-.0)
xEN
0<0<1
1
1-0
0
(1- 0)2
(1 - 0) exp(t)
1-0exp(t)
No. of trials uutil (and including)
first failure
e - probability of success
Hypergeometric
HyGe( n, N, M)
(~)(~=~)
(~)
xE {max{O,n-(N -MH,
N,nEN
ME {O, . .. ,N}
nO
N-n
nO(1 - 0) . - N-l
_3
No. of type I objects in a sample of
size n, drawn without replacement
from a population of size N, containing M type I objects.
xE{k,k+l, ... }
kEN
0<0<1
k
1-0
kO
(1- 0)2
((1 - 0) exp(t)) k
1- Oexp(t)
No. of trials uutil (and including)
ktlt failure
e - probability of success
NeBi(l, e) == Geo(e)
x ENo
,>,>0
,>,
,>,
exp(,>,(exp(t) -1))
Negative Binomial
NeBi(k,O)
e-
Poisson
POi('>')
,>,X
exp(-,>,)x!
... ,min{n,M}}
l)ox-k(l_ ol
k-l
(withe =
#)
Bi(n, e) can be approximated by Poi(ne), if n large, e small and ne moderate.
2 Bi(n,
3
Comments
function M(t)
(p.m.f.) p(x)
e) can be approximated by N(ne, ne(l- e», ifn large and e not too close to 0 or l.
No simple closed form expression exists.
Continuous univariate distributions
Distribution
Range
Probability density function
Parameters
Var(X)
JE(X)
Moment-generating
Beta
Be(al, a2)
x(};I-I(I- x)(};2- I
1
Cauchy
Ca(11, 'Y)
1r'Y
Chi-Squared
x~-Iexp(-~)
Exponential
Expo(fJ)
F
F(VI, V2)
0::; x::; 1
B(al,a2)
X 2 (v)
xElR
(1 + (x~~)2)
2~r (~)
Oexp( -fJx)
~
v!'
B
~
al
a2
>0
>0
11 E lR
(VIX
+ V2)2
Vl +V2
al
al +a2
aIa2
(al + a2)2(al + a2
+ 1)
_3
_4
_4
_4
1
(1 - 2t)~
x>O
vEN
V
2v
x>O
0>0
1
0
1
fJ2
x>O
Vb
V2
V2 - 2
2v~ (VI + v2 - 2)
VI(V2 - 2)2(V2 - 4)
v2 EN
(for 1/2
> 2)
(for 1/2
Xl ~ Ga(O:l' e)
X2 ~ Ga(0:2, e) independent
=> XI~X2 ~ Be(0:1,0<2)
'Y>O
v
xT-I
v 22
(!a..2 ' ~)
2
Comments
function M(t)
(p.d.f.) f(x)
Ca(O, 1)
Xi
=>
~
== t(l)
N(O, 1) independent
I:¥=l Xf ~ x 2(1/)
1
1-~
> 4)
_4
Xl ~ X 2 (1/l)
X2 ~ X2(1/2) independent
=> Xl/VI ~ F(1/l 1/2)
X2/ V2
'
(p.d'!.)
Normal
N(IL, ( 2 )
1
(
---exp
V21fa 2
x>O
(X- IL )2)
2a 2
xEIR
8k()
Comments
1
!±!
1
a~x~b
b-a
a
8
82
(1-
ILEIR
a2 > 0
IL
a2
exp ( ILt
it
2
a t2)
+ -2-
N(O, 1) - standard normal
x:;1-! '" N(O, 1)
8k 2
8k
8-1
(for () > 1)
_3
(8 - 1)2(8 - 2)
(for() > 2)
Xl ~ N(O, 1)
X2 ~ X 2 (v) independent
V
0
vEN
xEIR
..;vB (~, ~)(1 + X;) 2
a
a>O
8>0
Ga(1,8) == Expo(8)
Ga (~,~) == X 2(v)
1
8>0
k>O
x>k
X()+l
Uniform
U(a, b)
c.dJ. for a
Moment-generating
function M(t)
8<>X<>-1 exp( -8x)
r(a)
Student's t
t(v)
Var(X)
E(X)
Parameters
f(x)
Gamma
Ga(a,8)
Pareto
Pa(k,8)
(for v> 1)
_4
v-2
=?
(for v> 2)
a, bE IR
a<b
a+b
(b - a)2
2
12
a>O
r (1 + ~)
8>0
8f;
./fj ~ t(v)
exp(bt) - exp(at)
t(b - a)
U(O, 1)
_3
We(1,8)
== 8e(1, 1)
< x < b:
Weibull
a8x<>-lexp( -8x<»
We(a, 8)
3
Range
Probability density function
Distribution
x>O
r (1 ~ ~) _ (1E(X)f
== Expo(8)
No simple closed form expression exists.
4 Does not exist.
Multivariate distributions
Probability mass I density function
Distribution
Range
Parameters
E(Xj)
Var(Xj
)
Cov(Xi,Xj)
MultinOlnial
Mu(n, 81, .. . 8 k )
Normal
Nk(JL, :E)
n!__
xl 1.... Xk!
1
8Xl
I
...
8%k
~~=IXj =n
nEN
0< 8j < 1
~~=l 8j = 1
(1
x E IRk
JL E IRk
:E symm., pos. def.
1
21fva~a~(1
x ENo
..- - exp -"2(x - JL )T :E- I (x - JL) )
Special case: bivariate normal distribution with x
=(
Xl ) , JL
X2
Moment-generating function
M(t) = M(tl, ... , tk)
p(x) = p(XI"'" Xk) or f(x) = f(xl, ... , Xk)
=(
ILl ) , :E
1L2
(. a~(xI - ILI)2 - 2paW2(xI -ILI)(X2 -1L2)
exp 2 2(
2)
- p2)
2a l a 2 1- p
= ( a~a
p la2
+ a~(x2 -1L2)2)
n8j
n8 j (1- 8 j )
-n8i 8 j
(t,o;exp(t;)) "
ILj
r,jj
r,ij
exp (JL T t
ILj
a~
pa la2
pa la2 ).
.
a~
ILl, 1L2 E IR
a~,a~ > 0
-1 < p < 1
J
+ ~t T:Et)