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BernoulliDistribution
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BernoulliDistribution

✖

represents a Bernoulli distribution with probability parameter p.

Details

Background & Context

  • BernoulliDistribution[p] represents a discrete statistical distribution defined on the real numbers, where the parameter p is represents a probability parameter satisfying . The Bernoulli distribution is sometimes referred to as the coin toss distribution or as the distribution of a Bernoulli trial. It has a discrete probability density function (PDF) that returns the value p at , gives at , and evaluates to 0 for all other real numbers.
  • The Bernoulli distribution is named for Swiss mathematician Jacob Bernoulli and is designed to model the simple action of flipping a (fair or unfair) coin. Traditionally, p is thought of as the probability with which the experiment "succeeds" (so that 1 represents a successful experiment), whereas is the probability of "failure" (so that 0 represents a failed experiment). In the coin flip analogy, 1 typically represents heads, while tails is represented by 0. The value corresponds to flipping a fair coin. Despite being defined very simply, the Bernoulli distribution serves as the basis for a number of other, often more complicated mathematical concepts including the Bernoulli sequence in probability, the Bernoulli measure in measure theory, and the Bernoulli scheme in dynamical systems. Within the study of stochastic processes, the Bernoulli distribution is also the motivation behind the so-called Bernoulli process (BernoulliProcess), a discrete-time stochastic process consisting of a (finite or infinite) sequence of random variables, each of which is independent and identically Bernoulli distributed. Moreover, many real-world scenarios showing a well-defined dichotomy of independent outcome possibilities can be modeled as Bernoulli processes. Examples include the probability of rolling a particular value with a single (fair) die and the numbers of defective products, given a defect rate that is independent of the production scale.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision pseudorandom variates from a Bernoulli distribution. Distributed[x,BernoulliDistribution[p]], written more concisely as xBernoulliDistribution[p], can be used to assert that a random variable x is distributed according to a Bernoulli distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[BernoulliDistribution[p],x] and CDF[BernoulliDistribution[p],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
  • DistributionFitTest can be used to test if a given dataset is consistent with a Bernoulli distribution, EstimatedDistribution to estimate a Bernoulli parametric distribution from given data, and FindDistributionParameters to fit data to a Bernoulli distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Bernoulli distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Bernoulli distribution.
  • TransformedDistribution can be used to represent a transformed Bernoulli distribution and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Bernoulli distribution and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Bernoulli distributions.
  • BernoulliDistribution is related to a number of other probability distributions. For example, BernoulliDistribution[p] is equivalent to a single instance of BinomialDistribution[1,p], i.e. PDF[BernoulliDistribution[p],k] is identical to Piecewise[Table[{PDF[BinomialDistribution[1,p],l],kl},{l,0,1}]]. Similarly, the sum of n independent Bernoulli variables with common success rate p is modeled by BinomialDistribution[n,p]. In addition, a number of naturally occurring quantities emerging from collections of independent Bernoulli-distributed random variables may be modeled according to other well-known distributions. For example, the number of successes in the first n data points distributed according to BernoulliDistribution[p] has distribution BinomialDistribution[n,p] while the number of trials to get one (respectively r) successes has distribution GeometricDistribution[p] (respectively NegativeBinomialDistribution[r,p]).

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability mass function:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-eul0y3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-c01mzx
Out[2]=2

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-dgrh1v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-emx029
Out[2]=2

Mean and variance:

In[1]:=1
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-klg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-e21
Out[2]=2

Median:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-1nmdo9
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a Bernoulli distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-6ywy61

Compare the frequency of 1 in the sample with the probability of getting 1:

In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-yl2jw2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-nd9nuv
Out[3]=3

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-epi747
Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-f8ui5o
Out[3]=3

Skewness:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-9f7swl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-u39
Out[2]=2

The distribution is symmetric for :

In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-tk4hd
Out[3]=3

Kurtosis:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-vg6wag
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-og0
Out[2]=2

Find where the kurtosis attains its minimum:

In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-3cb4z2
Out[3]=3

Different moments with closed forms as functions of parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-f4yfjm

Moment:

In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-xqhud7
Out[2]=2

CentralMoment:

In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-5udluv
Out[3]=3

Closed form for symbolic order:

In[4]:=4
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-mp1ffr
Out[4]=4

FactorialMoment:

In[5]:=5
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-z5zkug
Out[5]=5

Cumulant:

In[6]:=6
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-kshtug
Out[6]=6

Hazard function:

In[1]:=1
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-1ogkwy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-f3dt10
Out[2]=2

Quantile function:

In[1]:=1
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-dkj9r3
In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-eic3y2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-knd5o2
Out[3]=3

Use dimensionless Quantity to define BernoulliDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-f619wx
Out[1]=1

Applications  (6)Sample problems that can be solved with this function

Simulate a sequence of fair coin tosses:

In[1]:=1
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-iftl6f
Out[1]=1

The probability of throwing a 6 on a six-sided die can be modeled as a Bernoulli distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-eoi4ve
Out[1]=1

Simulate throwing a die, if you are only interested in sixes:

In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-j7rf9h
Out[2]=2

Out of 10 bulbs produced, one is defective. Simulate production of 100 bulbs:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-jgqnhr
Out[1]=1

Find the number of good bulbs in the batch:

In[2]:=2
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https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-21od0k
Out[2]=2

Find the expected number of good bulbs per batch of 100:

In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-ff8rws
Out[3]=3

Find the probability that a randomly selected bulb is good:

In[4]:=4
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-ss7fsf
Out[4]=4

A lottery sells 10 tickets for $1 per ticket. Each time there is only one winning ticket. A gambler has $5 to spend. Find his probability of winning if he buys 5 tickets in 5 different lotteries:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-s51k7a
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-bvbb39
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-c48k9f
Out[3]=3

His probability of winning is greater if he buys 5 tickets in the same lottery:

In[4]:=4
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-ylgh68
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-mqgisf
Out[5]=5

Simulate a symmetric random walk with values -1 and 1:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-nv1zec
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-yvfkcb
Out[2]=2

In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of Poisson distribution and another distribution, depending on the type of light. If the source uses coherent laser light of intensity , then the electron count distribution is Poisson:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-uqf6l4
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-dic0qt
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-tt31wm
In[4]:=4
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-pei7m
Out[4]=4

Which is PoissonDistribution:

In[5]:=5
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-1deqea
Out[5]=5

If the source uses thermal illumination, then the Poisson parameter follows ExponentialDistribution with parameter and the electron count distribution is:

In[6]:=6
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-i5fej4
Out[6]=6

These two distributions are distinguishable and allow the type of source to be determined:

In[7]:=7
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-6ig422
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

The probability of getting anything other than zero and one is zero:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-uc2
Out[1]=1

Relationships to other distributions:

BernoulliDistribution is equivalent to BinomialDistribution of one trial:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-b6jbet
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-6a8g32
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-zrxsty
Out[3]=3

BinomialDistribution is the sum of independent Bernoulli variables:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-vvfb5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-c54dur
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

BernoulliDistribution is not defined when p is not between zero and one:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-yta
Out[1]=1

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

In[1]:=1
✖
https://wolfram.com/xid/0bsxo3qzvpb9tiyn4a-vmb
Out[1]=1
Wolfram Research (2007), BernoulliDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliDistribution.html (updated 2016).
✖
Wolfram Research (2007), BernoulliDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliDistribution.html (updated 2016).

Text

Wolfram Research (2007), BernoulliDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliDistribution.html (updated 2016).

✖
Wolfram Research (2007), BernoulliDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliDistribution.html (updated 2016).

CMS

Wolfram Language. 2007. "BernoulliDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BernoulliDistribution.html.

✖
Wolfram Language. 2007. "BernoulliDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BernoulliDistribution.html.

APA

Wolfram Language. (2007). BernoulliDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliDistribution.html

✖
Wolfram Language. (2007). BernoulliDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliDistribution.html

BibTeX

@misc{reference.wolfram_2025_bernoullidistribution, author="Wolfram Research", title="{BernoulliDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BernoulliDistribution.html}", note=[Accessed: 13-March-2025 ]}

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@misc{reference.wolfram_2025_bernoullidistribution, author="Wolfram Research", title="{BernoulliDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BernoulliDistribution.html}", note=[Accessed: 13-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_bernoullidistribution, organization={Wolfram Research}, title={BernoulliDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BernoulliDistribution.html}, note=[Accessed: 13-March-2025 ]}

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@online{reference.wolfram_2025_bernoullidistribution, organization={Wolfram Research}, title={BernoulliDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BernoulliDistribution.html}, note=[Accessed: 13-March-2025 ]}