Check the parameter convention for a negative binomial waiting count

Define the task

A waiting-count interpretation stops when a target number r of successes has occurred and records the preceding failures. State this endpoint explicitly. Some count-regression interfaces instead describe a negative binomial distribution through its mean and dispersion. Matching a parameter name without checking its definition can change the distribution.

Carry out the check

For independent attempts with constant success probability p, mean failures are r(1−p)/p and failure-count variance is r(1−p)/p². Adding r converts failures to total attempts, changing the mean but leaving the variance unchanged. The literal stopping interpretation requires a positive integer target. Mathematical extensions allow a noninteger shape without representing a literal target success count.

Worked hypothetical example

For hypothetical r=3 and p=0.5, expected failures are three and variance is six. Expected total attempts are six. A report giving mean six and variance six could be describing total attempts, while mean three and variance six describes failures. Reconcile the endpoint before calling either table inconsistent. A varying success probability or dependent attempts requires a different sampling model.

Method checked 11 November 2026 against R documentation: The Negative Binomial Distribution.

Open the target-success waiting moments.