Bernoulli distribution

Bernoulli
	Probability mass function;
	Cumulative distribution function;
Parameters	(real)
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median	N/A
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

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In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability $p$ and value 0 with failure probability $q = 1 - p$ . So if X is a random variable with this distribution, we have:

$\Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!$

The probability mass function f of this distribution is

$f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.$

The expected value of a Bernoulli random variable X is $E\left(X\right)=p$ , and its variance is

$\textrm{var}\left(X\right)=p\left(1-p\right).\,$

The kurtosis goes to infinity for high and low values of p, but for $p = 1 / 2$ the Bernoulli distribution has a lower kurtosis than any other probability distribution, namely -2.

The Bernoulli distribution is a member of the exponential family.

Related distributions

If $X_1,\dots,X_n$ are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then $Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)$ (binomial distribution).
The Categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
The Beta distribution is the conjugate prior of the Bernoulli distribution.

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

v • d • e Statistics
Descriptive statistics	Mean (Arithmetic, Geometric) - Median - Mode - Power - Variance - Standard deviation
Inferential statistics	Hypothesis testing - Significance - Null hypothesis/Alternate hypothesis - Error - Z-test - Student's t-test - Maximum likelihood - Standard score/Z score - P-value - Analysis of variance
Survival analysis	Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models
Probability distributions	Normal (bell curve) - Poisson - Bernoulli
Correlation	Confounding variable - Pearson product-moment correlation coefficient - Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient)
Regression analysis	Linear regression - Nonlinear regression - Logistic regression

Bernoulli distribution

Related distributions

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Acknowledgement and Attribution Regarding Sources of Content

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Probability mass function
Cumulative distribution function
Parameters	$p>0\,$ (real)
Support	$k=\{0,1\}\,$
Probability mass function (pmf)	$\begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}$
Mean	$p\,$
Median	N/A
Mode	$\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}$
Variance	$pq\,$
Skewness	$\frac{q-p}{\sqrt{pq}}$
Excess kurtosis	$\frac{6p^2-6p+1}{p(1-p)}$
Entropy	$-q\ln(q)-p\ln(p)\,$
Moment-generating function (mgf)	$q+pe^t\,$
Characteristic function	$q+pe^{it}\,$