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Preliminary Course

Applied Mathematics

Probabilities. Negative Binomial Law


Negative Binomial Law

    Introductive Example
    Let us consider again stem growth.
    Assume that every month, there is a probability of 0.5 that a phytomer develops.
    Now, consider that, as a botanist, you are observing the stem at a given time t without knowing anything about the follow-up of the growth until that time, and want to make measurements on the last phytomer at the tip of that stem.

    Problem: you do not know when that last phytomer appeared.
    It might have appeared just recently (at the current month), or one month before and the stem paused during the current month (with probability 0.5).
    It can even happen that there were two successive pauses and that the phytomer is in fact 3 months-old. Or be even older than that if the stem had, by (bad) chance, undergone several successive pauses.

    To get information on its real chronological age, we need to assess the expected number of pauses that the stem undergoes before it creates a phytomer.
    So we need to study the number of trials (months) that it takes to get one success (phytomer appearance), and more generally, r successes.

    Consider a succession of Bernoulli trials and let us inquire how many trials it will take for the rth success to turn up.
    It will obviously take at least r trials.
    The probability that the rth success occurs at the trial number k (where k = r, r+1, ... ) equals the probability of a sequence consisting of exactly k - r failures and r successes: this occurs with probability (1 - p)k-rpr (all the trials are assumed to be independent).
    Since the rth success comes last, it remains to choose the positions of r - 1 other successes out of the remaining k - 1 trials.
    Hence the following definition


    Definition
    In a sequence of independent Bernoulli (p) trials, let the random variable X denote the trial at which the rth success occurs, where r is a fixed integer.
    Then X follows the negative binomial distribution NB(r, p) and:

      

    The negative binomial distribution is sometimes defined in terms of the random variable Y = number of failures before the rth success:

      

    This formulation is equivalent to the one above since Y = X-r.

    Some Properties

         Mean: E = E(NB(r,p)) = r. (1 - p) / p
         Variance: V = V(NB(r,p) = r . (1-p) / p2
         Therefore for both Binomial and Negative Binomial law, we have p. E = V

    The negative binomial distribution gets its name from the alternative (and equivalent) definition:

      
Definition

Negative Binomial Distribution
Mathematics. In the theory of probability and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number of failures occurs.

Exercise
    In the above introductive example

      What is the probability that the last phytomer (at stem tip) is in fact 3 months-old ?    
      and 1 month-old (ie has just appeared) ?         ??

      What is the expectation of the age of the last phytomer ?             ??