Chapter 3 Reference — Binomial • Geometric • Negative Binomial • Poisson

Clean PMF/CDF, calculators, and step-by-step derivations of mean and variance. Built to be a long-term study companion.

Binomial Distribution — Binomial(n, p)

Fixed number of independent trials n, success probability p each trial; X = number of successes.

PMF: P(X = k) = C(n,k) · p^k · (1−p)^n−k, k = 0,1,…,n

CDF: P(X ≤ k) = Σ_i=0..k C(n,i) pⁱ (1−p)ⁿ⁻ⁱ

Mean: μ = n p Variance: σ² = n p (1−p)

Quick calculator

n p k –

Excel: PMF =BINOM.DIST(k,n,p,FALSE) · CDF =BINOM.DIST(k,n,p,TRUE) · Legacy PMF =BINOMDIST(k,n,p,FALSE)

Deriving the mean and variance

Model X as a sum of n Bernoulli trials: Let X = X₁ + … + X_n, where each X_i is 1 if trial i is success, else 0.
Mean: E[X_i] = p, so E[X] = Σ E[X_i] = n p.
Variance: Var(X_i) = p(1−p); trials are independent, so variances add: Var(X) = Σ Var(X_i) = n p (1−p).

This argument avoids heavy algebra and is the standard, clean derivation.

Geometric Distribution — Geometric(p)

Independent, identical trials with success probability p. X = number of trials until the first success (support 1,2,…).

PMF: P(X = k) = (1−p)^k−1 · p, k = 1,2,…

CDF: P(X ≤ k) = 1 − (1−p)^k

Mean: μ = 1/p Variance: σ² = (1−p)/p²

Quick calculator

p k –

Excel: PMF =GEOM.DIST(k,p,FALSE) · CDF =GEOM.DIST(k,p,TRUE) · Legacy =GEOMDIST(k,p,FALSE/TRUE) · Fallback via NegBin: =NEGBINOM.DIST(k-1,1,p, FALSE/TRUE)

Deriving the mean and variance

Mean E[X]: Use the series identity Σ_k=1..∞ k r^k−1 = 1/(1−r)² for |r|<1. Here r = 1−p.

E[X] = Σ k (1−p)^k−1 p = p · Σ k r^k−1 = p · 1/(1−r)² = p · 1/p² = 1/p.
Variance: First get E[X²] using the identity Σ k² r^k−1 = (1+r)/(1−r)³ (obtained by differentiating the geometric series twice).

E[X²] = p · Σ k² r^k−1 = p · (1+r)/(1−r)³ = p · (2−p)/p³ = (2−p)/p².

Var(X) = E[X²] − (E[X])² = (2−p)/p² − 1/p² = (1−p)/p².

Equivalently, you can use the memoryless property to show E[X]=1/p and then compute Var(X) from series.

Negative Binomial — NegBin(r, p)

Independent, identical trials with success probability p. X = number of trials to accumulate r successes. (Geometric is r=1.)

PMF: P(X = k) = C(k−1, r−1) · (1−p)^k−r · p^r, k = r, r+1,…

CDF: P(X ≤ k) = Σ_i=r..k C(i−1,r−1) (1−p)^i−r p^r

Mean: μ = r/p Variance: σ² = r(1−p)/p²

Quick calculator

r p k –

Excel: PMF =NEGBINOM.DIST(k-r, r, p, FALSE) · CDF =NEGBINOM.DIST(k-r, r, p, TRUE)

Deriving the mean and variance

Key idea: The total trials to get r successes equals the sum of r independent geometric waiting times (each counts trials to the next success).
Mean: If G ~ Geometric(p), then E[G]=1/p. For independent G₁,…,G_r, X = G₁+…+G_r ⇒ E[X] = r · (1/p) = r/p.
Variance: Var(G)=(1−p)/p² and variances add (independence), so Var(X)= r (1−p)/p².

This “sum of geometrics” view is the cleanest derivation and matches the PMF given above.

Poisson — Poisson(λ)

Counts events in a fixed interval when events occur independently at a constant average rate λ.

PMF: P(X = x) = e^−λ · λ^x / x!, x = 0,1,2,…

CDF: P(X ≤ x) = Σ_k=0..x e^−λ λ^k / k!

Mean: μ = λ Variance: σ² = λ

Quick calculator

λ k –

Excel: PMF =POISSON.DIST(k, lambda, FALSE) · CDF =POISSON.DIST(k, lambda, TRUE) · Legacy: =POISSON(k, lambda, FALSE/TRUE)

Deriving the mean and variance

Mean: Start with E[X] = Σ x · e^−λ λ^x / x!. Shift index: write x = (x−1)+1 or factor one λ:

E[X] = e^−λ Σ_x≥1 x λ^x/x! = e^−λ Σ_x≥1 λ · λ^x−1/(x−1)! = λ · e^−λ Σ_y≥0 λ^y/y! = λ
Second factorial moment: E[X(X−1)] = e^−λ Σ x(x−1) λ^x/x! = e^−λ Σ λ² λ^x−2/(x−2)! = λ².
Variance: E[X²] = E[X(X−1)] + E[X] = λ² + λ. Hence Var(X) = E[X²] − (E[X])² = (λ²+λ) − λ² = λ.

These index-shift tricks avoid calculus and make Poisson’s μ=σ²=λ pop right out.