The negative binomial 1 waiting for success

In the next three sections, we will discuss the negative binomial distribution, which is perhaps one of the most versatile probability distributions used in ecology and evolutionary biology. There are two quite different derivations of the negative binomial distribution. The first, which we will do in this section, is relatively simple. The second, which requires an entire section of preparation, is more complicated, but we will do that one too.

Imagine that we are conducting a series of Bernoulli trials in which the probability of a success is p Rather than specifying the number of trials, we ask the question: how long do we have to wait before the kth success occurs? That is, we define a random variable N according to

Pr{N = n|k,p} = Probability that the kth success occurs on trial n (3.48)

Now, for the kth success to occur on trial n, we must have k — 1 successes in the first n — 1 trials and a success on the nth trial. The probability of k — 1 successes in n — 1 trials has a binomial distribution with parameters n — 1 and p and the probability of success on the nth trial has probability p and these are independent of each other. We thus conclude

PrfN = n|k, p}= ( n — ^ 1(1 — p)"^ = (j, — 1) / (1 — p)n—k

This is the first form of the negative binomial distribution.

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