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Binomial characteristics vs negative binomial

Short answer

Binomial: fixed n independent trials with constant p, counts successes; mean np, variance npq. Negative binomial: number of trials until k-th success, mean k/p, variance kq/p^2. The key difference: binomial fixes trials and counts successes; negative binomial fixes successes and counts trials.

What are the characteristics of Binomial Distribution and how does it differ from Negative Binomial Distribution?

Characteristics of the binomial distribution

  1. The experiment consists of a fixed number n of trials.
  2. Each trial has only two outcomes, success (S) or failure (F).
  3. The probability of success p is the same in every trial; q = 1 − p.
  4. The trials are independent.
  5. The random variable X counts the successes, so X = 0, 1, …, n.

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, mean = np, variance = npq (always less than the mean).

Negative binomial distribution

In repeated independent trials with constant p, let X be the trial on which the k-th success occurs:

P(X = x) = C(x − 1, k − 1) pᵏ qˣ⁻ᵏ, x = k, k + 1, …; mean = k/p, variance = kq/p².

Difference

BasisBinomialNegative binomial
Number of trialsFixed (n)Random
Number of successesRandomFixed (k)
X measuresSuccesses in n trialsTrials until k-th success
Possible values0, 1, …, nk, k+1, … (unbounded)
Meannpk/p
Variancenpqkq/p²

Example: the number of heads in 10 tosses is binomial; the number of tosses needed to get 3 heads is negative binomial.

✓ Answer checked by Claude Sonnet 5.5

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