Definition:Bernoulli Distribution
Definition
Let $X$ be a discrete random variable on a probability space.
Then $X$ has the Bernoulli distribution with parameter $p$ if and only if:
- $(1): \quad X$ has exactly two possible values, for example $\Img X = \set {a, b}$
- $(2): \quad \map \Pr {X = a} = p$
- $(3): \quad \map \Pr {X = b} = 1 - p$
where $0 \le p \le 1$.
That is, the probability mass function is given by:
- $\map {p_X} x = \begin {cases} p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \set {a, b} \\ \end {cases}$
If we allow:
- $\Img X = \set {0, 1}$
then we can write:
- $\map {p_X} x = p^x \paren {1 - p}^{1 - x}$
Success or Failure
The actual values of $a$ and $b$ depends on the particular experiment in question.
However, it is conventional to consider that the outcome whose probability is $p$ is determined to be a success, while the other outcome is determined to be a failure.
Also defined as
In the definition of a Bernoulli distribution, some sources insist that $0 < p < 1$.
However, it can be useful in certain circumstances to include the condition when the outcome is certainty.
Notation
The Bernoulli distribution can be written:
- $X \sim \Bernoulli p$
but as, from Bernoulli Process as Binomial Distribution, the Bernoulli distribution is the same as the binomial distribution where $n = 1$, the notation:
- $X \sim \Binomial 1 p$
is often preferred, for notational economy.
Frequently $q$ is used for $1 - p$ in which case the probability mass function is given by:
- $\map {p_X} x = \begin {cases} p & : x = a \\ q & : x = b \\ 0 & : x \notin \set {a, b} \\ \end {cases}$
where $p + q = 1$.
Also see
- Results about the Bernoulli distribution can be found here.
Source of Name
This entry was named for Jacob Bernoulli.
Technical Note
The $\LaTeX$ code for \(\Bernoulli {p}\) is \Bernoulli {p} .
When the argument is a single character, it is usual to omit the braces:
\Bernoulli p
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples: $(6)$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bernoulli distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bernoulli distribution