Bernoulli distribution

Key facts

Notation

B e r n o u l l i ( p ) {\displaystyle \mathrm {Bernoulli} (p)}

Parameters

0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} , q = 1 − p {\displaystyle q=1-p}

Support

k ∈ { 0 , 1 } {\displaystyle k\in \{0,1\}}

Pmf

{ q = 1 − p if k = 0 p if k = 1 {\displaystyle {\begin{cases}q=1-p&{\text{if }}k=0\\p&{\text{if }}k=1\end{cases}}}

Cdf

{ 0 if k < 0 1 − p if 0 ≤ k < 1 1 if k ≥ 1 {\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

p {\displaystyle p}

Median

1/2\n \\end{cases}"}}'> { 0 if p < 1 / 2 [ 0 , 1 ] if p = 1 / 2 1 if p > 1 / 2 {\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

1/2\n \\end{cases}"}}'> { 0 if p < 1 / 2 0 , 1 if p = 1 / 2 1 if p > 1 / 2 {\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

p ( 1 − p ) = p q {\displaystyle p(1-p)=pq}

Mad

2 p ( 1 − p ) = 2 p q {\displaystyle 2p(1-p)=2pq}

Skewness

q − p p q {\displaystyle {\frac {q-p}{\sqrt {pq}}}}

Excess kurtosis

1 − 6 p q p q {\displaystyle {\frac {1-6pq}{pq}}}

Entropy

− q ln ⁡ q − p ln ⁡ p {\displaystyle -q\ln q-p\ln p}

Mgf

q + p e t {\displaystyle q+pe^{t}}

Cf

q + p e i t {\displaystyle q+pe^{it}}

Pgf

q + p z {\displaystyle q+pz}

Fisher information

1 p q {\displaystyle {\frac {1}{pq}}}

via Wikipedia infobox

Wikidata facts

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