Statistics 100B
University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou Probability Distributions – Summary
Discrete Distributions
Distribution
Probability Mass Function
Mean
Variance
Moment-generating Function
Binomial
P(X = x) = npx(1 − p)n−x x
x = 0,1,···,n
np
np(1 − p)
[pet + (1 − p)]n
Geometric
P(X = x) = (1 − p)x−1p x = 1,2,···
1
p
1−p p2
pet 1−(1−p)et
Negative Binomial
P(X = x) = x−1pr(1 − p)x−r r−1
x = r, r + 1, · · ·
r p
r(1−p) p2
[pet ]r 1−(1−p)et
Hypergeometric
(r)(N−r) P(X=x)= x n−x
( Nn )
x = 0, 1, · · · , n if n ≤ r,
x = 0, 1, · · · , r if n > r
nr N
nr N−rN−n N N N−1
Fairly complicated!
Poisson
P(X = x) = λxe−λ
x! x = 0,1,···
λ
λ
exp[λ(et − 1)]
Continuous Distributions
Distribution
Probability Density Function
Mean
Variance
Moment-generating Function
Uniform
f(x)= 1 b−a
a≤x≤b
a+b 2
(b−a)2 12
etb −eta t(b−a)
Gamma
−x
f(x)=xα−1e β,α,β>0,x≥0 βαΓ(α)
αβ
αβ2
(1 − βt)−α
Exponential
f(x) = λe−λx, λ > 0, x ≥ 0
1
λ
1
λ2
(1 − 1 t)−1 λ
Beta
f(x) = xα−1(1−x)β−1 B(α,β)
α>0, β>0, 0≤x≤1
α α+β
αβ (α+β)2(α+β+1)
Normal
f(x)= 1 e−1(x−μ)2 √2σ
σ 2π −∞ < x < +∞
μ
σ2
eμt+t2σ2 2
Remarks:
Binomial: Geometric: Negative Binomial: Hypergeometric: Poisson:
X represents the number of successes among n trials.
X represents the number of trials needed until the first success.
X represents the number of trials needed until r successes occur.
X represents the number of items among the n selected that comes from the r group. X represents the number of events that occur in time, area, etc.