In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the Bernoulli distribution, named after Swiss mathematician
Jacob Bernoulli,
[James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45] is the
discrete probability distribution of a
random variable which takes the value 1 with probability
and the value 0 with probability
. Less formally, it can be thought of as a model for the set of possible outcomes of any single
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
that asks a
yes–no question. Such questions lead to
outcomes that are
boolean-valued: a single
bit whose value is success/
yes
Yes or YES may refer to:
* An affirmative particle in the English language; see yes and no
Education
* YES Prep Public Schools, Houston, Texas, US
* YES (Your Extraordinary Saturday), a learning program from the Minnesota Institute for Talent ...
/
true/
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
with
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
''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
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
is a random variable with this distribution, then:
:
The
probability mass function of this distribution, over possible outcomes ''k'', is
:
This can also be expressed as
:
or as
:
The Bernoulli distribution is a special case of the
binomial distribution with
The
kurtosis goes to infinity for high and low values of
but for
the two-point distributions including the Bernoulli distribution have a lower
excess kurtosis than any other probability distribution, namely −2.
The Bernoulli distributions for
form an
exponential family.
The
maximum likelihood estimator of
based on a random sample is the
sample mean.
Mean
The
expected value of a Bernoulli random variable
is
:
This is due to the fact that for a Bernoulli distributed random variable
with
and
we find
:
Variance
The
variance of a Bernoulli distributed
is
:
We first find
:
From this follows
:
With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside