# Negative Binomial Distribution definition, formula, properties with applications

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### Definition

A discrete random variable X is said to have negative binomial distribution if its probability density function is defined as,

where r>0 and 0<=p<=1 are the two parameter of the distribution such that p+q=1. f(x;r,p) is the probability of getting exactly r successes in (x+r) independent bernoulli trials.

### Properties of Negative Binomial Distribution

• The mean of negative binomial distribution is .
• The variance of the distribution is.
• The skewness of the distributionis.
• The kurtosis of negative binomial distribution is .

### Characteristics of Negative Binomial Distribution

Some characteristics of  negative binomial is given below-
• If r=1, negative binomial distribution tends to geometric distribution with parameter p.
• If r➝∞, q➝0 and rq➝λ, then negative binomial distribution tends to poission distribution.
• The variance of negative binomial distribution is always greater than mean.
• The distribution is positively skewed and leptokurtic.

### Applications of Negative Binomial Distribution

There are some real life application of negative binomial distribution-
• The negative binomial distribution is applicable in those data set where variance is greater than mean.
• When poission unable to describe a data set or inadequate then we prefer negative binomial distribution.
• Used in accident statistics ( birth and death process).
• Used in psychological data set.
• Also used in time series data in economics, medical and military data etc.