Number of failures until \(k\)'th success
Confusingly, there is an alternative definition of a negative binomial random variable that is also in common use. We defined the variable on the previous page as the number of trials until the \(k\)'th success, but some other authors define \(X\) to be the number of failures until the \(k\)'th success is observed. We will use \(X^*\) to denote this random variable,
Definition
In a sequence of independent Bernoulli trials with \(P(success) = \pi > 0\) in each trial, the number of failures observed before the \(k\)'th success is observed has a distribution that is also called a negative binomial distribution.
\[ X^* \;\; \sim \; \; \NegBinDistn^*(k, \pi) \]If the \(k\)'th success is observed on the \(X\)'th trial, there must have been \((X - k)\) failures, so
\[ X^* \;\; = \; \; X - k \]Alternative distribution's probability function
If \( X^* \sim \NegBinDistn^*(k, \pi) \), then it has probability function
\[ p(x) = \begin{cases} \displaystyle{{x + k -1} \choose {k-1}} \pi^k(1-\pi)^x & \quad \text{for } x = 0, 1, \dots \\[0.5em] 0 & \quad \text{otherwise} \end{cases} \]The probability function follows directly from replacing \(x\) in the earlier probability function for \(X\) with \((x+k)\),
\[ P(X^* = x) = P(X = x+k) \]In answering questions relating to negative binomial distributions, it is important to carefully identify which type of negative binomial distribution should be used.