# Definition:Binomial Distribution

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## 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 $p$** if and only if:

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

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

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

- Results about
**the Binomial distribution**can be found here.

## Technical Note

The $\LaTeX$ code for \(\Binomial {n} {p}\) is `\Binomial {n} {p}`

.

When the arguments are single characters, it is usual to omit the braces:

`\Binomial n p`

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.2$: Examples: $(7)$

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 39$: Probability Distributions: Binomial Distribution: $39.1$