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 o ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the Bernoulli distribution, named after Swiss mathematician
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
,
[James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45] is the
discrete probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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 Causality, cause-and-effect by demonstrating what outcome oc ...
that asks a
yes–no question
In linguistics, a yes–no question, also known as a binary question, a polar question, or a general question is a question whose expected answer is one of two choices, one that provides an affirmative answer to the question versus one that provid ...
. Such questions lead to
outcomes that are
boolean-valued: a single
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
whose value is success/
yes/
true
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* ...
/
one with
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
''p'' and failure/no/
false/
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
with probability ''q''. It can be used to represent a (possibly biased)
coin toss
A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to ...
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
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
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
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
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
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
with
The
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
goes to infinity for high and low values of
but for
the two-point distributions including the Bernoulli distribution have a lower
excess kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
than any other probability distribution, namely −2.
The Bernoulli distributions for
form an
exponential family
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
.
The
maximum likelihood estimator
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statis ...
of
based on a random sample is the
sample mean
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...
.
Mean
The
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
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