Bernoulli distribution
discrete probability distribution which compels the random variable to take one of two values
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability
p
{\displaystyle p}
and the value 0 with probability
q
=
1
−
p
{\displaystyle q=1-p}
. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are Boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have
p
≠
1
/
2.
{\displaystyle p\neq 1/2.}
The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.
Properties
If
X
{\displaystyle X}
is a random variable with a Bernoulli distribution, then:
Pr
(
X
=
1
)
=
p
,
Pr
(
X
=
0
)
=
q
=
1
−
p
.
{\displaystyle {\begin{aligned}\Pr(X{=}1)&=p,\\\Pr(X{=}0)&=q=1-p.\end{aligned}}}
The probability mass function
f
{\displaystyle f}
of this distribution, over possible outcomes k, is
f
(
k
;
p
)
=
{
p
if
k
=
1
,
q
=
1
−
p
if
k
=
0.
{\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}}
This can also be expressed as
f
(
k
;
p
)
=
p
k
(
1
−
p
)
1
−
k
for
k
∈
{
0
,
1
}
{\displaystyle f(k;p)=p^{k}(1-p)^{1-k}\quad {\text{for }}k\in \{0,1\}}
or as
f
(
k
;
p
)
=
p
k
+
(
1
−
p
)
(
1
−
k
)
for
k
∈
{
0
,
1
}
.
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