## 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)= Âµ *= *E*(*X*) = *np*

The variance of the binomial distribution is ,

* Varience(X)=Ïƒ*^{2} = *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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