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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 p {\displaystyle p} and the value 0 with probability q = 1 − p {\displaystyle q=1-p} . Less formally, it can be thought of as a model for…

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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}}}
CF
q + p e i t {\displaystyle q+pe^{it}}
Entropy
− q ln ⁡ q − p ln ⁡ p {\displaystyle -q\ln q-p\ln p}
Excess kurtosis
1 − 6 p q p q {\displaystyle {\frac {1-6pq}{pq}}}
Fisher information
1 p q {\displaystyle {\frac {1}{pq}}}
MAD
2 p ( 1 − p ) = 2 p q {\displaystyle 2p(1-p)=2pq}

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Properties

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Entropy and Fisher's Information

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Bernoulli distribution

Nodes48
Edges47
Triples39
Avg. degree1.96
Density0.041667
Components1

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Bernoulli distribution

Top relations

related to External links · 9
Bernoulli distribution → Binomial, EMS Press, Encyclopedia, Eric, Interactive, Mathematics, MathWorld, Univariate Distribution Relationships, Weisstein
related to Related distributions · 5
Bernoulli distribution → Bernoulli, If, Rademacher, The, The Beta
related to Fisher's Information · 3
Bernoulli distribution → Bernoulli, Fisher, For
related to Properties · 3
Bernoulli distribution → Bernoulli, If, Pr
is a · 2
Bernoulli distribution → special case of the binomial distribution where a single trial is conducted, special case of the binomial distribution with n
CDF · 1
Bernoulli distribution → { 0 if k 0 1 − p if 0 ≤ k 1 1 if k ≥ 1 {\displaystyle {\begin{cases}0&{\text{if }}k0\\1-p&{\text{if }}0\leq k1\\1&{\text{if }}k\geq 1\end{cases}}}
CF · 1
Bernoulli distribution → q + p e i t {\displaystyle q+pe^{it}}
Entropy · 1
Bernoulli distribution → − q ln ⁡ q − p ln ⁡ p {\displaystyle -q\ln q-p\ln p}
Excess kurtosis · 1
Bernoulli distribution → 1 − 6 p q p q {\displaystyle {\frac {1-6pq}{pq}}}
Fisher information · 1
Bernoulli distribution → 1 p q {\displaystyle {\frac {1}{pq}}}

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Important terminology

displaystyle bernoulli distribution probability 1-p random begin end outcomes frac entropy pr aligned variable pq value information operatorname single would

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Bernoulli distributionCDF{ 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}}}1.00infobox
Bernoulli distributionCFq + p e i t {\displaystyle q+pe^{it}}1.00infobox
Bernoulli distributionEntropy− q ln ⁡ q − p ln ⁡ p {\displaystyle -q\ln q-p\ln p}1.00infobox
Bernoulli distributionExcess kurtosis1 − 6 p q p q {\displaystyle {\frac {1-6pq}{pq}}}1.00infobox
Bernoulli distributionFisher information1 p q {\displaystyle {\frac {1}{pq}}}1.00infobox
Bernoulli distributionMAD2 p ( 1 − p ) = 2 p q {\displaystyle 2p(1-p)=2pq}1.00infobox
Bernoulli distributionMeanp {\displaystyle p}1.00infobox
Bernoulli distributionMedian{ 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}}}1.00infobox
Bernoulli distributionMGFq + p e t {\displaystyle q+pe^{t}}1.00infobox
Bernoulli distributionMode{ 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}}}1.00infobox
Bernoulli distributionNotationB e r n o u l l i ( p ) {\displaystyle \mathrm {Bernoulli} (p)}1.00infobox
Bernoulli distributionParameters0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} q = 1 − p {\displaystyle q=1-p}1.00infobox
Bernoulli distributionPGFq + p z {\displaystyle q+pz}1.00infobox
Bernoulli distributionPMF{ 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}}}1.00infobox
Bernoulli distributionSkewnessq − p p q {\displaystyle {\frac {q-p}{\sqrt {pq}}}}1.00infobox
Bernoulli distributionSupportk ∈ { 0 , 1 } {\displaystyle k\in \{0,1\}}1.00infobox
Bernoulli distributionVariancep ( 1 − p ) = p q {\displaystyle p(1-p)=pq}1.00infobox
Bernoulli distributionis aspecial case of the binomial distribution where a single trial is conducted0.90text
Bernoulli distributionis aspecial case of the binomial distribution with n0.90text

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