(3) Geometric and Negative Binomial Distribution

P(X=x) for 1st success of geometric distribution

P(X=x) = p(1-p)^x-1

Cumulative geometric distribution formula

P(X≤x) = 1-(1-p)^x

P(X≥x) = (1-p)^x-1

E(X) of geometric distribution

E(X) = μ = 1/p

Var(X) of geometric distribution

Var(X) = σ² = (1-p)/p²

P(X=x) for number of trials needed for r success of negative binomial distribution

P(X=x) = (x-1)C(r-1) p^r (1-p)^x-r

P(X≤a) for NB where X~NB(r,p)

Change X~NB(r,p) to Y~B(a,p)

Calculate P(Y≥r) and vice versa for P(X≥a)

E(X) of negative binomial distribution

E(X) = μ = r/p

Var(X) of negative binomial distribution

Var(X) = σ² = r(1-p)/p²