Definition:Binomial Distribution

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

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


 * $\Img X = \set {0, 1, \ldots, n}$


 * $\map \Pr {X = k} = \dbinom n k p^k \paren {1 - p}^{n - k}$

where $0 \le p \le 1$.

Note that this distribution gives rise to a probability mass function satisfying $\map \Pr \Omega = 1$, because:
 * $\displaystyle \sum_{k \mathop \in \Z} \dbinom n k p^k \paren {1 - p}^{n - k} = \paren {p + \paren {1 - p} }^n = 1$

This is apparent from the Binomial Theorem.

It is written:
 * $X \sim \Binomial n p$

Also defined as
Some sources insist that $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.

Also see

 * Binomial Distribution PMF
 * Bernoulli Process as Binomial Distribution