These variables are not multinomially distributed because their sum " X " 1 + & + " X " " m " is not fixed, being a draw from a negative binomial distribution.
32.
If we are tossing a coin, then the negative binomial distribution can give the number of heads ( " success " ) we are likely to encounter before we encounter a certain number of tails ( " failure " ).
33.
When counting the number of " " trials given " " r " " failures, the expected total number of trials of a negative binomial distribution with parameters is \ frac { r } { 1-p }.
34.
The Poisson distribution, the stuttering Poisson distribution, the negative binomial distribution, and the Gamma distribution are examples of infinitely divisible distributions as are the normal distribution, Cauchy distribution and all other members of the stable distribution family.
35.
If " r " is a counting number, the coin tosses show that the count of successes before the " r " th failure follows a negative binomial distribution with parameters " r " and " p ".
36.
If we used the negative binomial distribution to model the number of goal attempts a sportsman makes before scoring " r " goals, though, then each unsuccessful attempt would be a " success ", and scoring a goal would be " failure ".
37.
Clearly, P ? in this Poisson model for randomly distributed individuals is also the SPO . Other probability distributions, such as the negative binomial distribution, can also be applied for describing the SPO and the occupancy-abundance relationship for non-randomly distributed individuals.
38.
Thus, the negative binomial distribution is equivalent to a Poisson distribution with mean " pT ", where the random variate " T " is gamma-distributed with shape parameter " r " and intensity 1-" p ".
39.
The definition of negative binomial that I was taught and said above is different to the one on WP so from now on I'm talking about the one defined in the Negative binomial distribution article : The number of failures before r successes in an infinite series of Bernoulli ( p ) trials.
40.
In probability theory and statistics, the "'negative binomial distribution "'is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified ( non-random ) number of failures ( denoted " r " ) occurs.
