Negative Binomial Calculator

Negative Binomial Calculator – Two Definitions, One Flexible Distribution

The negative binomial distribution is a flexible discrete distribution for modeling repeated Bernoulli trials. Instead of fixing the number of trials and counting successes as in the binomial distribution, you fix the number of required successes r and let the random variable track either the number of failures before the r-th success or the trial on which the r-th success occurs. This calculator on MyTimeCalculator supports both parameterizations and brings the main formulas together in one place.

Throughout the article, p denotes the probability of success on each trial and q = 1 − p denotes the probability of failure. All trials are assumed to be independent with the same p from trial to trial.

Definition 1: Failures Before the r-th Success

In the failures parameterization the random variable X counts how many failures occur before the r-th success. The possible values are k = 0, 1, 2, … and we write X ∼ NB(r, p). The probability mass function is

PMF (failures parameterization)

P(X = k) = C(k + r − 1, k) p^r q^k, for k = 0, 1, 2, …

where C(k + r − 1, k) is the binomial coefficient “k + r − 1 choose k”.

CDF and tail probabilities

The cumulative distribution function and right-tail probabilities are given by finite sums of the PMF:

• P(X ≤ k) = Σ_{j = 0}^k C(j + r − 1, j) p^r q^j
• P(X ≥ k) = Σ_{j = k}^∞ C(j + r − 1, j) p^r q^j = 1 − P(X ≤ k − 1)

Mean and variance for failures

For this parameterization the mean and variance are

• E[X] = r q / p
• Var(X) = r q / p²

As p increases the expected number of failures before the r-th success decreases. As p decreases and q increases, the distribution becomes more spread out and the variance grows.

Definition 2: Trial of the r-th Success

In the trial-count parameterization the random variable T counts the total trial index on which the r-th success occurs. The possible values are t = r, r + 1, r + 2, … and the PMF is

PMF (trial-count parameterization)

P(T = t) = C(t − 1, r − 1) p^r q^{t − r}, for t = r, r + 1, …

This formula reflects that the r-th success must happen on trial t, with r − 1 successes scattered among the first t − 1 trials and the remaining t − r trials being failures.

Relationship between X and T

The two parameterizations are linked by a simple shift:

• T = X + r
• X = T − r

Substituting k = t − r into the failures PMF recovers the trial-count PMF. Many properties, such as the variance, are shared between the two views because shifting a distribution by a constant does not change its variance.

Mean and variance for T

Using the relationship T = X + r and the formulas for X, the mean and variance of T become

• E[T] = r / p
• Var(T) = r q / p²

The variance matches that of X while the mean shifts upward by r because T includes the r successful trials themselves.

Connection to the Geometric Distribution

The geometric distribution is the special case r = 1. In the failures parameterization, when r = 1 the PMF reduces to

P(X = k) = q^k p, for k = 0, 1, 2, …

which is exactly the distribution of the number of failures before the first success. The mean and variance collapse to the geometric formulas E[X] = q / p and Var(X) = q / p². This shows that the negative binomial distribution is a natural extension of the geometric distribution to the case of r required successes.

Connection to the Binomial Distribution

The binomial distribution chooses a fixed number of trials n and counts the number of successes. The negative binomial chooses a fixed number of required successes r and effectively lets the number of trials be random instead. The combinatorial term C(k + r − 1, k) mirrors the C(n, k) term in the binomial PMF, but the roles of the parameters are swapped.

Moment Generating Function (MGF) and Probability Generating Function (PGF)

For the failures parameterization X ∼ NB(r, p) the moment generating function, when it exists, is

MGF of X

M_X(t) = [p / (1 − q e^t)]^r, for values of t such that |q e^t| < 1.

The probability generating function is

PGF of X

G_X(z) = [p / (1 − q z)]^r, for |z| < 1 / q.

Derivatives of the PGF at z = 1 and derivatives of the MGF at t = 0 recover the mean and variance formulas already summarized above. These functions are useful in more advanced probability theory, branching processes and count data modeling.

Using the Negative Binomial Calculator

The calculator is structured to match these theoretical ideas:

• Failures tab: You enter r, p and a particular failure count k. The calculator evaluates P(X = k) = C(k + r − 1, k) p^r q^k, the CDF P(X ≤ k) and the tail probability P(X ≥ k), along with the mean r q / p and variance r q / p².
• Trials tab: You enter r, p and a trial t ≥ r. The calculator evaluates P(T = t) = C(t − 1, r − 1) p^r q^{t − r}, the CDF P(T ≤ t) and the tail probability P(T ≥ t), along with the mean r / p and variance r q / p².
• Table tab: You set r, p and a maximum failure count k_max. The tool builds a probability table for k from 0 to k_max with PMF, CDF and tail probabilities side by side.
• Properties tab: You enter r and p and the calculator evaluates E[X], Var(X), E[T], Var(T) and lists the MGF and PGF formulas for reference.

All calculations are based directly on the standard formulas for the negative binomial distribution using the failures before r-th success parameterization as the primary reference and then mapping to the trial-count view when needed.

Practical Tips for Modeling with the Negative Binomial

• Use the failures parameterization when you naturally think in terms of “how many misses before hit number r”.
• Use the trial-count parameterization when you track the exact position of the r-th success in a sequence.
• Check that r is a positive integer and that p lies strictly between 0 and 1, since boundary values collapse the distribution.
• Remember that heavy tails can appear for small p, because most of the mass shifts to larger counts of failures or trials.
• In count regression or overdispersed Poisson modeling, interpret r as a shape parameter that controls extra-Poisson variability.

This Negative Binomial Calculator is intended for learning, coursework, planning and quick probability checks. For complex applied work and model fitting, use it alongside full statistical software or professional guidance.