Bernoulli distribution

Bernoulli

Parameters	${\displaystyle 0<p<1,p\in \mathbb {R} }$
Support	${\displaystyle k\in \{0,1\}\,}$
pmf	${\displaystyle {\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}}$
CDF	${\displaystyle {\begin{cases}0&{\text{for }}k<0\\1-p&{\text{for }}0\leq k<1\\1&{\text{for }}k\geq 1\end{cases}}}$
Mean	${\displaystyle p\,}$
Median	${\displaystyle {\begin{cases}0&{\text{if }}q>p\\0.5&{\text{if }}q=p\\1&{\text{if }}q<p\end{cases}}}$
Mode	${\displaystyle {\begin{cases}0&{\text{if }}q>p\\0,1&{\text{if }}q=p\\1&{\text{if }}q<p\end{cases}}}$
Variance	${\displaystyle p(1-p)(=pq)\,}$
Skewness	${\displaystyle {\frac {1-2p}{\sqrt {pq}}}}$
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	${\displaystyle {\frac {1}{p(1-p)}}}$

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes the value 1 with probability ${\displaystyle p}$ and the value 0 with probability ${\displaystyle q=1-p}$. It can be used to represent a coin toss where 1 and 0 would represent "head" and "tail" (or vice versa), respectively. In particular, unfair coins would have ${\displaystyle p\neq 0.5}$.

The Bernoulli distribution is a special case of the binomial distribution where a single experiment/trial is conducted (n=1). It is also a special case of the two-point distribution, for which the two possible outcomes need not be 0 and 1.

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