Get best statistics assignment help service. Send your Binomial Distribution problems online and receive A+ grade solution
“Get binomial distribution statistics assignment help online“
Binomial Distribution Definition
Binomial distribution is the discrete probability distribution of the number of successes in a sequence of n statistical independence that is yes/no experiments where each of which yields success with a probability of p . Such a success/failure experiment is also called a Bernoulli experiment and when n = 1 , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular Binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn from a population of size N and if the sampling is carried out without replacement, the draws are not independent. Then the resulting distribution is a Hypergeometric distribution and not a binomial one and for N much larger than n , the binomial distribution is a good approximation and so it is widely used.
Probability mass function
In general, if the random variable K follows the binomial distribution with parameters n and p , then it will be K ~ B(n, p) and the probability of getting exactly k successes in n trials is given by the probability mass function as follows
for k = 0, 1, 2, ..., n , where
is the Binomial coefficient "n choose k" which is also denoted as C(n, k), nCk, or nCk. Suppose it is required to have k successes (pk) and n − k failures (1 − p)n − k. The k successes can occur anywhere among the n trials. Also there are C(n, k) different ways of distributing k successes in a sequence of n trials.
In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values because for k > n/2 , the probability can be calculated by its complement as follows:
Considering the expression of ƒ(k, n, p) which is a function of k , there is a k value that maximizes it and this k value can be found by calculating as follows
and comparing it to 1, then there is always an integer M that satisfies
The ƒ(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M , with the exception of the case where (n + 1)p is an integer which have two values for which ƒ is maximal: (n + 1)p and (n + 1)p − 1.
