Definition:Bernoulli Trial
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Definition
A Bernoulli trial is an experiment whose sample space has two elements, which can be variously described, for example, as:
- Success and failure
- True and False
- $1$ and $0$
- the classic heads and tails.
Formally, a Bernoulli trial is modelled by a probability space $\struct {\Omega, \Sigma, \Pr}$ such that:
- $\Omega = \set {a, b}$
- $\Sigma = \powerset \Omega$
- $\map \Pr a = p, \map \Pr b = 1 - p$
where:
- $\powerset \Omega$ denotes the power set of $\Omega$
- $0 \le p \le 1$
That is, $\Pr$ obeys a Bernoulli distribution.
Bernoulli Variable
Let $X$ be a discrete random variable whose sample space is $\Omega$ in such a Bernoulli trial.
Then $X$ is known as a Bernoulli variable.
Also defined as
Some sources insist that the valid codomain of a Bernoulli trial is $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.
Also see
- Results about Bernoulli trials can be found here.
Source of Name
This entry was named for Jacob Bernoulli.
Historical Note
The concept of a Bernoulli trial was first raised by Jacob Bernoulli in $1713$.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Bernoulli trial
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bernoulli trial (Jacques Bernoulli)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bernoulli trial (Jacques Bernoulli, 1713)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bernoulli trial