Negative Binomial Distribution

Defination

The experiment in which the trials are repeated until a fixed number (trials) of sucesses occurs is called a negative binomial experiment.In probability theory and statistics the negative binomial distribution is also called a descrete probability distribution of a number of sucess in a sequence of independent and identical distributed Bernoulli trials before a specified (non-random) number of failure occurs.In such a experiment, we find the probability that at least n trials wii be required to get a specific number of k sucess i.e. we find the probability that k^thsucess occurs at x^thtrial. In other words, we find the probability that there are x failures preceding the k^th sucesses in x+k trials.Where x is the random variable which represents the number of trials needed to produce k sucesses is called a negative binomial random variable. Also this random variable X is the inverse of the binomial random variable. This is because in binomial distribution, the number of trials n is kept fixed and the number of sucess X is a random variable, but in negative binomial distribution the number of trials is a random variable and the number of sucesses is fixed. In such a case, a binomial distribution hsa a negative index. Therefore, a binomial distribution with a negative index is called negative binomial distribution.

Thus, a random variable X is said to follow a negative binomial distribution with a parameters k and p if its probability mass function (p.m.f) is

$$P( X = x ) \ = p (x) =\binom{ x + k - 1 }{ k- 1 }p^kq^x\; x = 0, 1, 2, ..., and \ k > 0$$

This distribution is also called waiting time binomial distribution or Pascal disrtibution due to the French Mathematician Blaise Pascal ( 1623 - 62 ).

Derivation Of Negative Binomial Distribution

Let us consider an experiment consists of n independent trials and having the probability p of sucesses is constant for each trials suppose in a binomial experiment. Suppose the number of failures is represented by X preceeding the k^th sucesses in x+k trials.Here the total number of trials ( n ) required to produce k sucesses is n = x + k. Where k represents the fixed number of sucesses required and X is a random variable ( r.v ). Now we need to find out the probability that k^thsuccess occurs in x + k trials, which is the probability that exactly x failures preceed the k^th success in x + k trials. Here, the last trial must be success with probability p. In remaining x + k - 1 trials, there are k - 1 success whose probability is $$\binom{ x + k - 1 }{ k-1 }$$p^k-1q^x.

Hence, from multiplication theorem of probability, the probability of x failures preceeding the k^th success in x + k trials is given by

$$p(x) \ =\binom{ x+k-1 }{ k-1 }p^kq^x .$$

$$p(x) \ =\binom{ x+k-1 }{ k-1}p^kq^x\; x = 0, 1, 2, .... and \ k > 0$$

we have,

$$\binom{x+k-1}{k-1} \ = \binom{x+k-1}{x}$$

= $$\frac{ ( x+k-1 ) ( x+k-2 ) ... ( k+1 )k }{x!}$$

= $$\frac{ ( -1 )^x( -k ) ( -k-1 ) ... ( -k-x+2 ) ( -k-x+1 )}{x!}$$

= ( -1 )^x$$\binom{ -k }{ x }$$

Therefore,$$p(x) \ = \binom{ -k }{ x} p^k(-q)^x\; x = 0, 1, 2, ... and \ k > 0$$

Which is the ( x+1 )^th term in the expression of p^k ( 1 - q )^-k, a binomial with negative index. This is the reason that this probability distribution is called negative binomial distribution.

We can also use the notation (symbol) X ~ B^-( k, p ) or NB ( k, p ) to denote the random variable X follows negative binomial distribution with k and p as parameter. The probability mass function ( p.m.f ) p ( x ) is also sometimes can be written as p( x; k, p ).

It can be verified that

$$\sum_{x=0}^\infty p( x ) \ = p^k\sum_{x=0}^\infty \binom{-k}{x}$$

= p^k ( 1- q )^-k ( from Taylorseries expansion)

= 1

Alternative form of negative binomial distribution

An alternative form of negative binomialdistribution i.e X ~ B^-( k, p ) can be expredssed as the probability that k^thsucsess occurs in x^thtrials. Since the k^thsuccess occurs in the last x^th trials whoose probability is p and the( k-1 ) success occurs in the remaining ( x-1 ) trials,$$with\ the \ probability \binom{x-1}{k-1} p^k-1 q^{(x-1)-(k-1 )}$$ .

Therefore, the probability of getting k^thsuccess in x^th trial is given by the product of the two probabilities as

p ( x) = $$\binom{x-1}{k-1}$$ p^k-1q^(x-1)-(k-1).p

= $$\binom{x-1}{k-1}p^k q^x-k ; x = k, k+1, ...

thus, the random variable (r.v) X is the total number of trials upto and incliuding the k^thsucess and this X has a negative binomial distribution with parameters k ( number of successes required ) and p ( probability of succeess at any trials ).Again, if we assume

p = $$\frac{1}{Q}$$ and q = $$\frac{p}{Q}$$ so that Q - P = 1 as p + q = 1, then

p (x) = $$\binom{-k}{x}$$ Q^-k ( $$\frac{-P}{Q}$$ )^x ;x = 0, 1, 2, 3, ...

which is the general term in the negative binomial expansion (Q - P)^-K.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.