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 $p$ 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: $(7)$
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- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 39$: Probability Distributions: Binomial Distribution: $39.1$
