. Distribution The following are the two types of Theoretical distributions : . Discrete distribution . Continous distribution Discrete distribution The binomial and Poisson distributions are the most useful theoretical distributions for discrete variables.
XII Std - Business Maths & Stat EM Chapter - - . . Binomial distribution Binomial distribution was discovered by James Bernoulli( - ) in the year and was first published posthumously in , eight years after his death. A random experiment whose outcomes are of two types namely success S and failure F, occurring with probabilities p and q respectively, is called a Bernoulli trial.
Some examples of Bernoulli trials are : (i) Tossing of a coin (Head or tail) (ii) Throwing of a die (getting even or odd number) Consider a set of n independent Bernoullian trails ( n being finite) in which the probability ‘ p ’ of success in any trial is constant , then q = –p , is the probability of failure. The probability of x successes and consequently ( n–x ) failures in n independent trials, in a specified order (say) SSFSFFFS….FSF is given in the compound probability theorem by the expression P(SSFSFFFS…..FSF) = P(S)P(S)P(F)P(S)… P(F)P(S)P(F) = p . p . q.p … q .
q . q ... = { x factors} {( n – x ) factors} = p q n x x successes in n trials can occur in n C x ways and the probability for each of these ways is same namely p q n x − . The probability distribution of the number of successes, so obtained is called the binomial probability distribution and the binomial expansion is ( q + p ) n Definition .
A random variable X is said to follow binomial distribution with parameter n and p , if it assumes only non- negative value and its probability mass function in given by P ( X = x ) = p ( x ) = n