Binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each which yields success with probability p. A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. The binomial distribution is often used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution. When N is much larger than n, the binomial distribution is a good approximation, and widely used.

Typically the mode of the binomial distribution is equal to [(n + 1)p] where [] is the floor function. However, (n+1)p is an integer and p is either 0 now 1, then the distribution has two mode, (n+1)p and (n+1)p-1. Where p is equal to 0 or 1, the mode will be 0 and n correspondingly. There is no single formula to find the median for a binomial distribution, and it may even be non-unique.

If n is large enough, then the skew of the distribution is not too great. In this case, a reasonable approximation to B(n, p) is given by the normal distribution. The baic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases and is better when p is not near to 0 to 1.

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with the parameter can be used as an approximation to B(n,p) of the binomial distribution if n is sufficiently large and p is sufficiently small.

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