### Binomial Distribution

This is also a random variable distribution. A random experiment is conducted n times independently. Suppose E is one event of the experiment, P is its probability. If each time the event occurs treat it as success otherwise treat it as failure then probability for success is equal to P.

Probability for failure (say) q = 1-p → p+q =1.

Now, out of the ‘n’ times number of success that we may get it 0 or 1 or 2 or…………..n

Probability of getting r times success =

The random variable distribution in the values of random variable are the number of success is called __Binomial Distribution__.

Now, the binomial distribution can be written as follow:

### Probability Distribution Function

#### Symbol :-

If x_{1}, x_{2}, x_{3},…….., x_{n} are values of a random variable X, then

for any two real number K_{1}, K_{2}.

for any real number K.

#### Definition :-

Probability distribution function of a random variable X is denoted by F and is defined on R as follow:

#### Properties :-

If F is an imaginary function, then

#### Theorems :-

- The arithmetic mean of binomial variable X with parameters n, p is the product of n and p ie., n.p | here n is the number of times that the experiment is conducted and p is the probability of success.
- Variance of a binomial variable X with parameters n, p is n.p.q where q = 1 – p