Definition:Bernoulli Distribution

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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, e.g. $\operatorname{Im} \left({X}\right) = \left\{{a, b}\right\}$
$(2): \quad \Pr \left({X = a}\right) = p$
$(3): \quad \Pr \left({X = b}\right) = 1 - p$

where $0 \le p \le 1$.


That is, the probability mass function is given by:

$p_X \left({x}\right) = \begin{cases} p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \left\{{a, b}\right\} \\ \end{cases}$


If we allow:

$\operatorname{Im} \left({X}\right) = \left\{{0, 1}\right\}$

then we can write:

$p_X \left({x}\right) = p^x \left({1-p}\right)^{1-x}$


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

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


Notation

This distribution is sometimes written:

$X \sim \operatorname{Bern} \left({p}\right)$

but as, from Bernoulli Process as Binomial Distribution, the Bernoulli distribution is the same as the binomial distribution where $n = 1$, the notation:

$X \sim \operatorname{B} \left({1, p}\right)$

is often preferred, for notational economy.


Frequently $q$ is used for $1-p$ in which case the probability mass function is given by:

$p_X \left({x}\right) = \begin{cases} p & : x = a \\ q & : x = b \\ 0 & : x \notin \left\{{a, b}\right\} \\ \end{cases}$

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.


Sources