Definition:Binomial Distribution

Definition
Let $$X$$ be a discrete random variable on a probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

Then $$X$$ has the binomial distribution with parameters $$n$$ and $$p$$ if:


 * $$\operatorname{Im} \left({X}\right) = \left\{{0, 1, \ldots, n}\right\}$$


 * $$\Pr \left({X = k}\right) = \binom n k p^k \left({1-p}\right)^{n-k}$$

where $$0 \le p \le 1$$.

Note that this distribution gives rise to a probability mass function satisfying $$\Pr \left({\Omega}\right) = 1$$, because:
 * $$\sum_{k \in \Z} \binom n k p^k \left({1-p}\right)^{n-k} = \left({p + \left({1-p}\right)}\right)^n = 1$$

This is apparent from the Binomial Theorem.

It is written:
 * $$X \sim \operatorname{B} \left({n, p}\right)$$

Also see

 * Binomial Distribution PMF
 * Bernoulli Process as Binomial Distribution