Bernoulli distribution information

Bernoulli distribution
	Probability mass function Three examples of Bernoulli distribution: and and and
Parameters	;
Support
PMF
CDF
Mean
Median
Mode
Variance
MAD
Skewness
Excess kurtosis
Entropy
MGF
CF
PGF
Fisher information

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,^[1] is the discrete probability distribution of a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q=1-p$ . 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 $p\neq 1/2.$

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. ^[2]

^ Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.
^ Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529.

[1] Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.

[2] Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529.

Bernoulli distribution information

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Bernoulli distribution

Binomial distribution

Continuous Bernoulli distribution

Categorical distribution

Bernoulli trial

Multinomial distribution

Beta distribution

Bernoulli process

Bernoulli family

Probability distribution

List of probability distributions

Joint probability distribution

List of things named after members of the Bernoulli family

Normal distribution

Poisson binomial distribution

Convolution of probability distributions

Prior probability

Jacob Bernoulli

Laplace distribution

Generalized linear model

Relationships among probability distributions

Kurtosis

Sufficient statistic

Markov chain

Logistic distribution

Rademacher distribution

Probability mass function

Probability mass function Three examples of Bernoulli distribution: $P(x=0)=0{.}2$ and $P(x=1)=0{.}8$ $P(x=0)=0{.}8$ and $P(x=1)=0{.}2$ $P(x=0)=0{.}5$ and $P(x=1)=0{.}5$
Parameters	$0\leq p\leq 1$ $q=1-p$
Support	$k\in \{0,1\}$
PMF	${\begin{cases}q=1-p&{\text{if }}k=0\\p&{\text{if }}k=1\end{cases}}$
CDF	${\begin{cases}0&{\text{if }}k<0\\1-p&{\text{if }}0\leq k<1\\1&{\text{if }}k\geq 1\end{cases}}$
Mean	$p$
Median	${\begin{cases}0&{\text{if }}p<1/2\\\left[0,1\right]&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}$
Mode	${\begin{cases}0&{\text{if }}p<1/2\\0,1&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}$
Variance	$p(1-p)=pq$
MAD	$2p(1-p)=2pq$
Skewness	${\frac {q-p}{\sqrt {pq}}}$
Excess kurtosis	${\frac {1-6pq}{pq}}$
Entropy	$-q\ln q-p\ln p$
MGF	$q+pe^{t}$
CF	$q+pe^{it}$
PGF	$q+pz$
Fisher information	${\frac {1}{pq}}$

Probability theory
Part of a series on statistics

Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
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