Binomial distribution

Binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

If the random variable X fulfils below with parameters n\inN and p\in[0,1], we say that X follows the binomial distribution B(n,p).

We write that as X \sim B(n,p).

    \[ P[X = k] = \binom{n}{k}p^k(1 - p)^{n-k}\; for\; k = 0,1,2,...,n \]

\binom{n}{k} is binomial coefficients.

(1)   \begin{eqnarray*} \binom{n}{k} = {}_n C _k = \frac{n!}{k!(n-k)!} \end{eqnarray*}

We call this distribution as Bernoulli distribution if n = 1

If X \sim B(n,p), expected value E[X] = np, variance V(X) = np(1-p)

    \[ E[X] = E[X_1 + X_2 + ... + X_n] = E[X_1] + E[X_2] + ... + E[X_n] = np \]

    \[ V[X] = V[X_1 + X_2 + ... + X_n] = V[X_1] + V[X_2] + ... + V[X_n] = n \left( p(1-p)^{2} + (1-p)p^{2} \right) = np\left(1-p\right) \]