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# Probability distributions

Source:
The Concise Oxford Dictionary of Mathematics

# Probability distributions

Discrete probability distributions

Distribution

Parameters

Probability mass function

Mean

Variance

Moment generating function

Binomial

n, p

r = 0, 1, 2, …, n

μ = np

σ2 = np(1 − p)

M(t) = (1 + p(et − 1))n

Geometric

p

P(X = r) = p(1 − p)r−1, r = 1, 2, …

Negative binomial

k, p

Hypergeometric

n, N, M

for r with max(0, nN + M)≤r≤min(n, M)

Not useful

Poisson

λ

r = 0, 1, 2, …

μ = λ

σ2 = λ

M(t) = exp{λ(et − 1)}

Continuous probability distributions

Distribution

Parameters

Probability density function

Mean

Variance

Moment generating function

Uniform on [a, b]

a, b

Normal

μ, σ

−∞ ≤ x ≤ ∞

μ

σ2

Exponential

λ

f(x)=λeλx, x ≥ 0

t<λ

Gamma

λ, r

t<λ

Chi−squared

n

μ=n

σ2=2n

, t<