The negative binomial distribution, also known as the Pascal distribution or Pólya distribution, gives the probability of r-1 successes and x failures in x+r-1 trials. Loi de Probabilite: Loi de Bernoulli, Loi Normale, Loi Uniforme Continue, Loi Binomiale Negative, Loi de Poisson, Loi de Student, Loi Norm. 5 likes. Ce. A propos de la repartition de la Cecidomyie du hetre Mikiola fagi.. Un exemple de distribution binomiale negative ().

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In probability theory and statisticsthe 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. A convention among engineers, climatologists, and others is to use “negative binomial” or “Pascal” for the case of an integer-valued stopping-time parameter rand use “Polya” for the real-valued case. For occurrences of “contagious” discrete events, like tornado outbreaks, the Polya distributions can be used to give more accurate models than the Poisson distribution by allowing the mean and variance to be different, unlike the Poisson.

Suppose there is a sequence of independent Bernoulli trials. Thus, each trial has two potential outcomes called “success” and “failure”. We are observing this sequence until a predefined number r of failures has occurred. Then the random number bunomiale successes we have seen, Xwill have the negative binomial or Pascal distribution:. When applied to real-world problems, outcomes of success and failure may or may not be outcomes we ordinarily view as good and bad, respectively.

Suppose we used the negative binomial distribution to model the number of days a certain machine works before it breaks down. In this case “success” would be the result on a day when the machine worked properly, whereas a breakdown would be a “failure”. If we used the negative binomial distribution to model the number of goal attempts an athlete makes before scoring r goals, though, then each unsuccessful attempt would be a “success”, and scoring a goal would be “failure”.

If we are tossing a coin, then negativf negative binomial distribution can give the number of heads “success” we are likely to encounter before we encounter a certain number of tails “failure”. The probability mass function of binomixle negative binomial distribution is. Here the quantity in parentheses is the binomial coefficientand is equal to.

This quantity can alternatively be written in the following manner, explaining the name “negative binomial”:. To see this intuitively, imagine the above experiment is performed many times.

That is, a biomiale of trials is performed until r failures are obtained, then another set of trials, and then another etc. Write down the number of trials performed in each experiment: Say the experiment was performed n times.

Negative Binomial Distribution

Negtaive there are nr failures in total. That is what we mean by “expectation”. This agrees with the mean given in the box on the right-hand side of this page. Negqtive sources may define the negative binomial distribution slightly differently from the primary one here.

The most common variations are where the random variable X is counting different things. Negaive variations can be seen in the table here:. Each of these definitions of the negative binomial distribution can be expressed in slightly different but equivalent ways. The first alternative formulation is simply an equivalent form of the binomial coefficient, that is: The second alternate formulation somewhat simplifies the expression by recognizing that the total number of trials is simply the number of successes and failures, that is: These second formulations may be more intuitive to binomale, however they are perhaps less practical as they have more terms.


Decrease of the aggregation parameter r towards zero corresponds to increasing aggregation of the organisms; increase of r towards infinity corresponds to absence of aggregation, as can be described by Poisson regression. For the special case where r is an integer, the negative binomial distribution is known as the Pascal distribution.

It is the probability distribution of a certain number of failures and successes in a series of independent and identically distributed Bernoulli trials. In other words, the negative binomial distribution is the probability distribution of the number of successes before the looi th failure in a Bernoulli processwith probability p of successes on each trial.

A Bernoulli process is a discrete time process, and so the number of trials, failures, and successes are integers. Consider the following example. Suppose we repeatedly throw a die, and consider a 1 to be a “failure”. That negagive of successes is a negative-binomially distributed random variable.

The negative binomial distribution, especially in its alternative parameterization described above, can be used as an alternative to the Poisson distribution. It is especially useful for discrete data over an unbounded positive range whose sample variance exceeds the sample mean.

In such cases, the observations are overdispersed with respect to a Poisson distribution, for which the mean is equal to the variance. Hence a Poisson lpi is not an appropriate model. Since the negative binomial distribution has one more parameter than the Poisson, the second parameter can be used to adjust the variance independently of the mean.

See Cumulants ibnomiale some discrete probability distributions.

Negative binomial distribution

An application of this is to annual counts negtaive tropical cyclones in the North Atlantic or to monthly to 6-monthly counts of wintertime extratropical cyclones over Europe, for which the variance is greater than the mean. The negative binomial distribution is also commonly used to model data in the form of discrete sequence read counts from high-throughput RNA and DNA sequencing experiments.

Consider a sequence of negative binomial random variables where the stopping parameter r goes to infinity, whereas the probability of success in each trial, pgoes to zero in such a way as to keep the mean of the distribution constant.

In other words, the alternatively parameterized negative binomial distribution converges to the Poisson distribution and r controls binomia,e deviation from the Poisson. This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Binomuale for large rbut which has larger variance than the Poisson for small r.

The negative binomial distribution also arises as a continuous mixture of Poisson distributions i. Together, the Success and Failure processes are equivalent to a single Poisson process of intensity 1, where an occurrence of the ndgative is a success if a corresponding independent coin toss comes up heads with probability p ; otherwise, it is a failure.

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. The count is also, however, the count of the Success Poisson process at the random time T of the r th occurrence in the Failure Poisson process. The following negatkve derivation which does not depend on r being a counting number confirms the intuition. Because of this, the negative binomial distribution is also known as negstive gamma—Poisson mixture distribution.

The negative binomial distribution was originally derived as a limiting case of the gamma-Poisson distribution. The negative binomial distribution is infinitely divisiblei. The negative binomial distribution NB rp can be represented as a compound Poisson distribution: Then the random sum. To prove this, we calculate the probability generating function G X of Xwhich is the composition of the probability generating functions G N and G Y 1. The following table describes four distributions related to the number of successes in a sequence of draws:.


The cumulative distribution function can be expressed in terms of the regularized incomplete beta function:. Suppose p is unknown and an experiment is conducted where it is decided ahead of time that sampling will continue until r successes are found. A sufficient statistic for the experiment is kthe number of failures. In estimating pthe minimum variance unbiased estimator is. The maximum likelihood estimate of p is. Suppose Y is a random variable with a binomial distribution with parameters n and p.

Then the binomial theorem implies that. Using Newton’s binomial theoremthis can equally be written as:. In this case, the binomial coefficient.

We can then say, for example. If r is a negative non-integer, so that the exponent is a positive non-integer, then some of the terms in the sum above are negative, so we do not binomiael a probability distribution on the set of all nonnegative integers.

Now we also allow non-integer values of r. Then we have a proper negative binomial distribution, which is a generalization of the Pascal distribution, which coincides with the Pascal distribution when r happens to be a positive integer. This property persists when the definition is thus generalized, and affords a quick way to see that the negative binomial distribution negativf infinitely divisible. The following recurrence relation holds:.

The maximum likelihood estimator only exists for samples for which the sample variance is larger than the sample mean. To find the maximum we take the partial derivatives with respect to r and p and set them equal to zero:.

This equation cannot be solved for r in closed form. If a numerical solution is desired, an iterative technique such as Newton’s method can be used. Alternatively, the expectation—maximization algorithm can be used. Pat Collis is required to sell candy bars to raise money for the 6th grade field trip. There are thirty houses in the neighborhood, and Pat is not supposed to return home until five candy bars have been sold.

So the child goes door to door, selling candy bars. At each house, there is a 0. Selling five candy bars means getting five successes. The number of trials i.

To finish on or before the eighth house, Pat must finish at the fifth, sixth, seventh, or eighth house. This can be expressed as the probability that Pat does not finish on the fifth through the thirtieth house:. Hospital length of stay is an example of real world data that can be modelled well with binomuale negative binomial distribution.

From Wikipedia, the free encyclopedia. Different texts adopt slightly different definitions for the negative binomial distribution. Probability and Statistics Second ed. The negative binomial distribution”. Negative Binomial Regression Second ed. Generalized Linear Models Second ed.

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Regression analysis of count data. Retrieved October 14, J R Stat Soc. Retrieved 23 April