Negative Binomial Distribution definition, properties with applications

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Definition

In a negative binomial experiment, a negative binomial random variable is the number X of repeated trials required to obtain r successes. A negative binomial distribution is the probability distribution of a negative binomial random variable. The Pascal distribution is another name for the negative binomial distribution. According to Wikipedia, The distribution of the number of trials required in an experiment to attain a specific number of successes was originally investigated in 1713 by Montmort. Pascal has previously mentioned this.

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

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

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.

Assume we repeatedly flip a coin and count the number of heads (successes). We’re performing a negative binomial experiment if we keep flipping the coin till it lands on heads two times. The number of coin flips required to acquire two heads is the negative binomial random variable. The number of coin flips is a random variable in this example, and it can have any integer value between 2 and infinity.

Properties of Negative Binomial Distribution

The mean of negative binomial distribution is ${frac {pr}{1-p}}$ .
The variance of the distribution is ${frac {pr}{(1-p)^{2}}}$ .
The skewness of the distribution is ${frac {1+p}{sqrt {pr}}}$ .
The kurtosis of negative binomial distribution is ${frac {6}{r}}+{frac {(1-p)^{2}}{pr}}$ .

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 Poisson 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 Poisson 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.

The negative binomial distribution is a specific instance of the geometric distribution. It is concerned with the number of trials needed to achieve a single accomplishment. As a result, the geometric distribution is a negative binomial distribution with one success rate (r).

Tossing a coin until it lands on heads is an example of a geometric distribution. What are the chances that the first head will appear on the third flip? The probability is known as a geometric probability and is denoted by the letter g. (x; P). The geometric probability formula is provided below.