8/17/2019 Probability Distributions Updated Final

http://slidepdf.com/reader/full/probability-distributions-updated-final 1/1

Discrete DistributionsName Notation Range PMF Expectation Variance Prob. Gen. Func.

Uniform(discrete)

  U  {m , . . . , n}   m, m + 1, . . . , n − 1, n  1

n−(m−1)m+n

2(n−(m−1))

2−112

s(n+1)−sm(n−(m−1))(s−1)

Bernoulli   Bern ( p) 0, 1

1   p;

0 1 − p. p p (1 − p)   1 − p + ps

Binomial   Bin (n, p) 0, 1, . . . , nnk

 pk (1 − p)n−k np np (1 − p)   (1 − p + ps)n

Geometric   G ( p) 1, 2, 3, . . .   p (1 − p)k−1   1 p

1− p p2

 ps1−(1− p)s

s <   11− p

NegativeBinomial

  NB (n, p)   n, n + 1, . . .k−1n−1

 pn (1 − p)k−n   n

 p

(1− p)n p2

  ps

1−(1− p)s

ns <   1

1− p

Hypergeometric   HG (N , D , n)

  max (0, n

−(N 

 −D)) ,

. . . , min (D, n)

(Dk)(N −Dn−k )

(N n)   n ·  D

N    n

D

N 

1 −  D

N  N −nN −1

  Too complex...

Poisson   P  (λ) 0, 1, 2, . . .  e−λλk

k!  λ λ   eλ(s−1)

Continuous DistributionsName Notation Range PDF Expectation Variance MGF

Uniform(continuous)

  U  (a, b)   (a, b)  1

b−aa+b

2(b−a)2

12

etb−eta

t(b−a)  , t = 0

1   , t = 0

Exponential   exp (λ)   (0,∞)   λe−λt   1λ

1λ2

λλ−t

, t < λ

Normal   N  (µ, σ)   (−∞,∞)  1√ 

2πσe−

  12σ2

(t−µ)2

µ σ2etµ+

1

2σ2t2

Gamma   Γ (s, λ)   (0,∞)  λs

Γ(s) ts−1e−λt   sλ

sλ2

  λλ−t

s, t < λ

Beta   B (a, b) (0, 1)  1

B(a,b)ta−1 (1 − t)b−1 a

a+bab

(a+b)2(a+b+1)   Too complex...

MultivariateNormal

  N 

µ, Σ

  Rn   1

(2π)n/2√ detΣ

exp−1

2

x − µ

T Σ−1

x − µ

  µ   Σ exp

tT  · µ +   1

2 tT Σt