Editing Bernoulli distribution (section)

{{Short description|Probability distribution modeling a coin toss which need not be fair}}
{{Use American English|date = January 2019}}
{{Probability distribution
|name       =Bernoulli distribution
|type       =mass
|pdf_image  =[[File:Bernoulli Distribution.PNG|325px|Funzione di densità di una variabile casuale normale]]
Three examples of Bernoulli distribution:
{{legend|7F0000|2=<math>P(x=0) = 0{.}2</math> and <math>P(x=1) = 0{.}8</math>}}
{{legend|00007F|2=<math>P(x=0) = 0{.}8</math> and <math>P(x=1) = 0{.}2</math>}}
{{legend|007F00|2=<math>P(x=0) = 0{.}5</math> and <math>P(x=1) = 0{.}5</math>}}
|cdf_image  =
|parameters =<math>0 \leq p \leq 1</math><br />
              <math>q = 1 - p</math>
|support    =<math>k \in \{0,1\}</math>
|pdf        =<math>\begin{cases}
    q=1-p & \text{if }k=0 \\
    p & \text{if }k=1
    \end{cases}
</math>
|cdf        =<math>\begin{cases}
    0 & \text{if } k < 0 \\
    1 - p & \text{if } 0 \leq k < 1 \\
    1 & \text{if } k \geq 1
    \end{cases}</math>
|mean       =<math> p</math>
|median     =<math>\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}</math>
|mode       =<math>\begin{cases}
    0 & \text{if } p < 1/2\\
    0, 1 & \text{if } p = 1/2\\
    1 & \text{if } p > 1/2
    \end{cases}</math>
|variance   =<math>p(1-p) = pq </math>
|mad        =<math>2p(1-p) = 2pq</math>
|skewness   =<math>\frac{q - p}{\sqrt{pq}}</math>
|kurtosis   =<math>\frac{1 - 6pq}{pq}</math>
|entropy    =<math>-q\ln q - p\ln p</math>
|mgf        =<math>q+pe^t</math>
|char       =<math>q+pe^{it}</math>
|pgf        =<math>q+pz</math>
|fisher     =<math> \frac{1}{pq} </math>|
}}
{{Probability fundamentals}}

In [[probability theory]] and [[statistics]], the '''Bernoulli distribution''', named after Swiss mathematician [[Jacob Bernoulli]],<ref>{{cite book |first=James Victor |last=Uspensky |title=Introduction to Mathematical Probability |publisher=McGraw-Hill |location=New York |year=1937 |page=45 |oclc=996937 }}</ref> is the [[discrete probability distribution]] of a [[random variable]] which takes the value 1 with probability <math>p</math> and the value 0 with probability <math>q = 1-p</math>. Less formally, it can be thought of as a model for the set of possible outcomes of any single [[experiment]] that asks a [[yes–no question]]. Such questions lead to [[outcome (probability)|outcomes]] that are [[Boolean-valued function|Boolean]]-valued: a single [[bit]] whose value is success/[[yes and no|yes]]/[[Truth value|true]]/[[Binary code|one]] with [[probability]] ''p'' and failure/no/[[false (logic)|false]]/[[Binary code|zero]] with probability ''q''. It can be used to represent a (possibly biased) [[coin toss]] where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails).  In particular, unfair coins would have <math>p \neq 1/2.</math>

The Bernoulli distribution is a special case of the [[binomial distribution]] where a single trial is conducted (so ''n'' would be 1 for such a binomial distribution). It is also a special case of the '''two-point distribution''', for which the possible outcomes need not be 0 and 1.<ref>{{cite book |last1=Dekking |first1=Frederik |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik |last4=Meester |first4=Ludolf |title=A Modern Introduction to Probability and Statistics |date=9 October 2010 |publisher=Springer London |isbn=9781849969529 |pages=43–48 |edition=1}}</ref>