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Discrete distribution; compare NB and binomial

Short answer

A discrete probability distribution assigns probabilities to each value of a discrete random variable, summing to 1, e.g., number of heads in two coin tosses. Binomial: fixed n trials, counts successes, mean np. Negative binomial: fixed k successes, counts trials until k-th success, mean k/p.

Define discrete probability distribution with suitable example. Compare Negative Binomial and Binomial probability distributions.

Discrete probability distribution

A discrete probability distribution lists every value x that a discrete random variable can take together with its probability P(X = x), called the probability mass function (pmf). It must satisfy:

  1. P(x) ≥ 0 for every x
  2. Σ P(x) = 1

Example: Toss a fair coin twice and let X = number of heads.

x012
P(x)1/41/21/4

All probabilities are non-negative and sum to 1, so this is a discrete distribution (it is binomial with n = 2, p = 0.5). Its mean is Σ x P(x) = 1.

Negative binomial vs binomial

  • Binomial: P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n.
  • Negative binomial: if X is the trial on which the k-th success occurs, P(X = x) = C(x − 1, k − 1) pᵏ qˣ⁻ᵏ, x = k, k + 1, …
BasisBinomialNegative binomial
Fixed quantityNumber of trials nNumber of successes k
Random variableNumber of successes in n trialsNumber of trials needed for k-th success
Range of X0 to n (finite)k, k+1, … (infinite)
Last trialAny outcomeMust be a success
Meannpk/p
Variancenpqkq/p²
Special casen = 1 gives Bernoullik = 1 gives geometric

Both need independent trials with constant p and two outcomes.

✓ Answer checked by Claude Sonnet 5.5

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