Negative Binomial Distribution definition, formula, properties with applications

Negative Binomial Distribution

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

$f\left&space;(&space;x;r,p&space;\right&space;)=\binom{n+r-1}{r-1}&space;p^{_{r}}q^{_{x}};&space;x=0,1,2,...,\infty&space;.$

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.