Appendix

Generic description: Number of Bernoulli trials until the first success Range of values: x = 1, 2, 3, . . . Parameter: p ∈ (0, 1), probability of success Probability mass function: P (x) = p(1− p)x−1 Cdf: 1− (1 − p)x

Expectation: μ = 1 p

Variance: σ2 = 1− p p2

Relations: Special case of Negative Binomial(k, p) when k = 1 A sum of n independent Geometric(p) variables is

Negative Binomial(n, p)

Negative Binomial(k, p)

Generic description: Number of Bernoulli trials until the k-th success Range of values: x = k, k + 1, k + 2, . . . Parameters: k = 1, 2, 3, . . ., number of successes

p ∈ (0, 1), probability of success Probability mass function: P (x) =

( x− 1 k − 1

) (1 − p)x−kpk

Expectation: μ = k

p

Variance: σ2 = k(1− p)

p2

Relations: Negative Binomial(k, p) is a sum of n independent Geometric(p) variables

Negative Binomial(1, p) = Geometric(p)

Poisson(λ)

Generic description: Number of “rare events” during a fixed time interval Range of values: x = 0, 1, 2, . . . Parameter: λ ∈ (0,∞), frequency of “rare events”

Probability mass function: P (x) = e−λ λx

x! Cdf: Table A3 Expectation: μ = λ Variance: σ2 = λ Relations: Limiting case of Binomial(n, p) when

n →∞, p → 0, np → λ Table: Appendix, Table A3

Beta(α, β)

Generic description: In a sample from Standard Uniform distribution, it is the distribution of the kth smallest observation

Range of values: 0 < x < 1 Parameter: α, β ∈ (0,∞), frequency of events, inverse scale parameter

Density: f(x) = Γ(α + β) Γ(α)Γ(β)

xα−1(1 − x)β−1

Expectation: μ = α/α + β Variance: σ2 = αβ/(α + β)2(α + β + 1)

Relations: As a prior distribution, Beta family is conjugate to the Binomial model.