Discrete Distributions

Count-based models for engineering data. Learn Bernoulli, binomial, geometric, negative binomial, Poisson and hypergeometric distributions — definitions, parameters, PMF, mean and variance, when to use each, engineering examples, and the Poisson approximation to the binomial.

How many defective chips in a lot? How many calls per minute at a server? How many trials until the first failure? Discrete distributions answer these counting questions.

An analogy

Binomial is flipping a biased coin n times; Poisson is counting raindrops on a tile per second; geometric is how many flips until the first head.

Choosing a model

Fixed number of independent trials, same p → binomial. Events in time/space at a constant rate → Poisson. Trials until first success → geometric (memoryless). Sampling without replacement from a small lot → hypergeometric.

Poisson approximation

Binomial with large n, small p (np moderate) ≈ Poisson(λ = np). Useful for rare defects and bit errors.

Acceptance sampling

Sampling plans accept a lot if ≤ c defects appear in a sample of n. The probability of acceptance vs lot quality is the operating characteristic (OC) curve.

Overdispersion

Real count data often have variance > mean (clustering); negative binomial models fit better than Poisson. Check assumptions before trusting the model.

Formulas

• Binomial: P(X = k) = C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ; μ = np, σ² = np(1 − p)
• Poisson: P(X = k) = e^(−λ)λᵏ/k!; μ = σ² = λ
• Geometric: P(X = k) = (1 − p)ᵏ⁻¹p; μ = 1/p
• Negative binomial: P(X = k) = C(k − 1, r − 1)pʳ(1 − p)ᵏ⁻ʳ; μ = r/p
• Hypergeometric: P(X = k) = C(K, k)C(N − K, n − k)/C(N, n)

Worked example: Defects in a sample

Given: n = 20 parts, p = 5 % defective

Find: P(0 defects), P(≤ 1).

• P(0) = 0.95²⁰ = 0.358
• P(1) = 20 × 0.05 × 0.95¹⁹ = 0.377
• P(≤ 1) = 0.736

Answer: 0.358, 0.736

Worked example: Server calls

Given: Average 3 calls/min (Poisson)

Find: P(5 calls), P(at least 1).

• P(5) = e⁻³3⁵/120 = 0.101
• P(≥ 1) = 1 − e⁻³ = 0.950

Answer: 0.101, 0.950

Common mistakes

• Wrong: Using binomial when sampling without replacement from a small lot. — Right: Use hypergeometric.
• Wrong: Poisson for non-constant rates. — Right: Rate must be constant over the interval.

Applications

• Quality control: Acceptance sampling plans.
• Networks: Poisson arrivals at servers and routers.
• Reliability testing: Trials until failure (geometric).

Key takeaways

• Binomial: fixed n, independent trials.
• Poisson: events at constant rate; μ = σ² = λ.
• Geometric: trials to first success.
• Hypergeometric: sampling without replacement.

Going further

• OC curves: Sampling-plan design.
• Poisson processes: Arrival models in queueing (M11).
• Zero-inflated models: Excess zeros in count data.