Binomial Distribution: Definition, Density function, properties and application

Binomial Distribution

 Let X be a discrete random variable, X is said to have binomial distribution if the density of X is defined  as,

     for r = 0, 1, 2, . . . , n.

Here , n = total number of observation

           r = number of trial

           p = probability of success

           q = probability of failure


So the probability distribution of X is called the binomial distribution. This is a discrete probability distribution.

Properties of a binomial distribution

  • Fixed number of trials, n, which means that the experiment is repeated a specific number of times.
  • The n trials are independent, which means that what happens on one trial does not influence the outcomes of other trials.
  • There are only two outcomes, which are called a success and a failure.
  • The probability of a success doesn’t change from trial to trial, where p = probability of success and q = probability of failure, q = 1 - p .

Mean and Variance of Binomial distribution

 The mean of the binomial distribution is the expectation of X which is as following,

                     Mean(X)= µ = E(X) = np

The variance of the binomial distribution is ,

                  Varience(X)=σ2 = V (X) = np(1 p) .

 Application of binomial distribution

In practical life we use binomial distribution when want to know the occurence of an event. For example-

  • Manufacturing company uses binomial distribution to detect the defective goods or items.
  • In clinical trail binomial trial is used to detect the effectiveness of the drug.
  • Moreover binomial trail is used in various field such as market research.

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