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:

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

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

where $0 \le p \le 1$.

Note that this distribution gives rise to a probability mass function satisfying $\Pr \left({\Omega}\right) = 1$, because:

$\displaystyle \sum_{k \in \Z} \binom n k p^k \left({1-p}\right)^{n-k} = \left({p + \left({1-p}\right)}\right)^n = 1$

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

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 2.2$