. . The Binomial Distribution The Binomial Distribution is an important distribution which applies in some cases for repeated independent trials where there are only two possible outcomes: heads or tails, success or failure, defective item or good item, or many other such possible pairs. The probability of each outcome can be calculated using the multiplication rule, perhaps with a tree diagram.
Suppose a coin is tossed once. Let X denote the number of heads. Then X Ber p ( ), because we get either head X or tail X with probability p or − p . Suppose a coin is tossed n times.
Let X denote the number of heads. Then X takes on the values , , , …, n . The probability for getting x number of heads is given by x p n x , x = , , , ..., n . - - X , corresponds to the combination of x heads in n tosses, that is n ways of heads and remaining n tails.
Hence, the probability for each of those outcomes is equal to p n x − Binomial theorem is suitable to apply when n is small number less than . Definition . : Binomial random variable A discrete random variable X is called binomial random variable, if X is the number of successes in n -repeated trials such that (i) the n- repeated trials are independent and n is finite (ii) each trial results only two possible outcomes, labelled as ‘success’ or ‘failure’ (iii) the probability of a success in each trial, denoted as p , remains constant. Definition .
: Binomial distribution The binomial random variable X equals the number of successes with probability p for a success and q for a failure in n- independent trials, has a binomial distribution denoted by X B n p ( , ). The probability mass function of X is x p n x , , ,..., . The name of the distribution is obtained from the binomial expansion . For constants a and b , the binomial expansion is x a b n x