Bernoulli distribution

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 ${\displaystyle p}$ and the value 0 with probability ${\displaystyle q=1-p}$. 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 ${\displaystyle p\neq 1/2.}$

Bernoulli distribution

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

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.