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 the binomial distribution gives rise to a probability mass function satisfying $\map \Pr \Omega = 1$, because:

$\ds \sum_{k \mathop \in \Z} \dbinom n k p^k \paren {1 - p}^{n - k} = \paren {p + \paren {1 - p} }^n = 1$

This is apparent from the Binomial Theorem.

It is written:

$X \sim \Binomial n p$

Also defined as

Some sources insist, in the definition of the binomial distribution, that $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.

Examples

Arbitrary Example

Let a die be cast $4$ times.

Let a score of $6$ be denoted as a success.

Then the experiment can be modelled by a binomial distribution $\Binomial n p$ such that $n = 4$ and $p = \dfrac 1 6$.

Thus the probability of $2$ successes is $\dfrac {25} {216}$.

Also see

• Probability Mass Function of Binomial Distribution
• Bernoulli Process as Binomial Distribution

• Definition:Negative Binomial Distribution

• 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: $(7)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): binomial distribution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial distribution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions

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