# 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,

$f(X=r;n,p)=\binom{n}{r}&space;p^{r}q^{1-r}&space;where&space;\binom{n}{r}=\frac{n!}{r!(n-r)!}$     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.