Negative Binomial Function

The negative binomial function is having the other name called Pascal distribution. Suppose we are having a sequence of independent trials we have two possible outcomes. One is success and another one is failure. Normally success is denoted as P and failure is denoted as q = 1 – P. we can observe from the sequence we will get r number of failures which is a predefined one. Then the number of success in random is we have seen X, will have the Negative binomial function. It is denoted like the following
                                               X `~~` NB (r, p)

Properties of Negative geometric Distribution:

              Let us take an independent trials we will have the probability for success is r – 1 and we have the probability for failure is x. so the total number of trials is x + r – 1. Then the success is on the (x +r) ^th trials.  Let us see the binomial distribution based on the coin problem. We can define these using two parameters. Where P is the probability of getting head and k is the number of heads where we want to stop the flipping. Here we will see some basic properties of negative binomial function.
Property 1:Probability of mass function:
            The probability of the coin tossing n times before obtaining k heads where P _k (L = n)
                                             P_k {L = n} = `[[n - 1],[k - 1]]`p^kq^{n - k}
               with n = k, k + 1, k + 2, . . .
                          and                  `[[A],[B]]` = `(A!)/(B!(A - B)!)`
              It is the number of combinations of B among A.
Property 2: Mean
              Mean µ = `k / p`
              In negative binomial function the mean is k times of the geometric distribution mean for the value p.
Property 3:Variance:
            Variance σ² = `kq / p^2`
            Variance of the Negative Binomial function is also k times of the variance of geometric distribution.

Relation between Negative Binomial variables and geometric variables:

              Let us take the geometric independent variable G_i , the sum of the geometric variables are
                                      L = `sum` _i G_i     where i = 1, 2, . . . . k
           The negative binomial distribution is having the parameter p with the size k. This is used to provide the simple definition of negative binomial function mean and variance.