Overview of Binomial Distribution

Trials of a random experiment are called Bernoulli trials if they satisfy the following

1. Each trial has exactly two outcomes: success or failure.

2. The probability of success remains the same in each trial.

The probability distribution of a random variable X, which represents the number of successes in n independent Bernoulli trials, each having probability of success p, is called the Binomial Distribution.

If q = 1 - p, then the probability function is given by 

\[P(X=x)=\binom{n}{x}p^xq^{n-x},\quad x=0,1,2,\ldots,n.\]

The probability of x successes, P(X = x), also denoted by P(x), is given by

\[P(x)=\binom{n}{x}q^{n-x}p^x,\quad x=0,1,2,\ldots,n,\quad(q=1-p).\]

This P(x) is called the probability function of the binomial distribution.

Var (X ) = npq

E(X) = np

\[\sigma=\sqrt{npq}\]