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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^{th}sucess occurs at x^{th}trial. 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

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

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

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 )

= ( -1 )

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

Which is the ( x+1 )

We can also use the notation (symbol) X ~ B

It can be verified that

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

= p

= 1

Therefore, the probability of getting k

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

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

thus, the random variable (r.v) X is the total number of trials upto and incliuding the k

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

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

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