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| | I'm Sherry and I live with my husband and our three children in Eberswalde, in the BB south part. My hobbies are Gardening, Martial arts and Baseball.<br><br>Feel free to surf to my website ... [http://fifa15-coingenerator.com/ FIFA coin generator] |
| {{Probability distribution|
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| name =Bernoulli|
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| type =mass|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>0<p<1, p\in\R</math>|
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| support =<math>k \in \{0,1\}\,</math>|
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| pdf =<math>
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| \begin{cases}
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| q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1
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| \end{cases}
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| </math>|
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| cdf =<math>
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| \begin{cases}
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| 0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1
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| \end{cases}
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| </math>|
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| mean =<math>p\,</math>|
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| median =<math>\begin{cases}
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| 0 & \text{if } q > p\\
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| 0.5 & \text{if } q=p\\
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| 1 & \text{if } q<p
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| \end{cases}</math>|
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| mode =<math>\begin{cases}
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| 0 & \text{if } q > p\\
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| 0, 1 & \text{if } q=p\\
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| 1 & \text{if } q < p
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| \end{cases}</math>|
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| variance =<math>p(1-p)\,</math>|
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| skewness =<math>\frac{q-p}{\sqrt{pq}}</math>|
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| kurtosis =<math>\frac{1-6pq}{pq}</math>|
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| entropy =<math>-q\ln(q)-p\ln(p)\,</math>|
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| mgf =<math>q+pe^t\,</math>|
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| char =<math>q+pe^{it}\,</math>|
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| pgf =<math>q+pz\,</math>|
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| fisher = <math> \frac{1}{p(1-p)} </math>|
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| }}
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| In [[probability theory]] and [[statistics]], the '''Bernoulli distribution''', named after Swiss scientist [[Jacob Bernoulli]], is a [[discrete probability distribution|discrete]] [[probability distribution]], which takes value 1 with success probability <math>p</math> and value 0 with failure probability <math>q=1-p</math>.
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| ==Properties==
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| If <math>X</math> is a random variable with this distribution, we have:
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| :<math> \Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!</math>
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| A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability <math>p</math> and tails with probability <math>1-p</math>. The experiment is called fair if <math>p=0.5</math>, indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).
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| The [[probability mass function]] <math>f</math> of this distribution is
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| : <math> f(k;p) = \begin{cases} p & \text{if }k=1, \\[6pt]
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| 1-p & \text {if }k=0.\end{cases}</math>
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| This can also be expressed as
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| :<math>f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}.</math>
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| The [[expected value]] of a Bernoulli random variable <math>X</math> is <math>E\left(X\right)=p</math>, and its [[variance]] is
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| :<math>\textrm{Var}\left(X\right)=p\left(1-p\right).\,</math>
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| Bernoulli distribution is a special case of the [[Binomial distribution]] with <math>n = 1</math>.<ref name="McCullagh1989Ch422">[[#McCullagh1989|McCullagh and Nelder (1989)]], Section 4.2.2.</ref>
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| The [[kurtosis]] goes to infinity for high and low values of <math>p</math>, but for <math>p=1/2</math> the Bernoulli distribution has a lower excess kurtosis than any other probability distribution, namely −2.
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| The Bernoulli distributions for <math>0 \le p \le 1</math> form an [[exponential family]].
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| The [[maximum likelihood estimator]] of <math>p</math> based on a random sample is the [[sample mean]].
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| ==Related distributions==
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| *If <math>X_1,\dots,X_n</math> are independent, identically distributed ([[i.i.d.]]) random variables, all Bernoulli distributed with success probability ''p'', then <math>Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)</math> ([[binomial distribution]]). The Bernoulli distribution is simply <math>\mathrm{Binomial}(1,p)</math>.
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| *The [[categorical distribution]] is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
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| *The [[Beta distribution]] is the [[conjugate prior]] of the Bernoulli distribution.
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| *The [[geometric distribution]] models the number of independent and identical Bernoulli trials needed to get one success.
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| ==See also==
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| *[[Bernoulli process]]
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| *[[Bernoulli sampling]]
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| *[[Bernoulli trial]]
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| *[[Binary entropy function]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * {{cite book | last = McCullagh | first = Peter | authorlink= Peter McCullagh | coauthors = [[John Nelder|Nelder, John]] | title = Generalized Linear Models, Second Edition | publisher = Boca Raton: Chapman and Hall/CRC | year = 1989 | isbn = 0-412-31760-5 |ref=McCullagh1989}}
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| *Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9
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| ==External links==
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| *{{springer|title=Binomial distribution|id=p/b016420}}
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| *{{MathWorld|title=Bernoulli Distribution|urlname=BernoulliDistribution}}
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| * Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]
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| {{ProbDistributions|discrete-finite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Bernoulli Distribution}}
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| [[Category:Discrete distributions]]
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| [[Category:Distributions with conjugate priors]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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I'm Sherry and I live with my husband and our three children in Eberswalde, in the BB south part. My hobbies are Gardening, Martial arts and Baseball.
Feel free to surf to my website ... FIFA coin generator