# BernoulliProcess

represents a Bernoulli process with event probability p.

# Examples

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## Basic Examples(3)

Simulate a Bernoulli process:

Mean and variance functions:

Covariance function:

## Scope(11)

### Basic Uses(5)

Simulate an ensemble of paths:

Compare paths for different values of the process parameter:

Process parameter estimation:

Estimate the distribution parameter from sample data:

Correlation function:

Absolute correlation function:

### Process Slice Properties(6)

Univariate SliceDistribution:

Univariate slice probability density:

Multi-time slice distribution:

Higher-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness does not depend on time:

The limiting values:

BernoulliProcess is symmetric for p=1/2:

Kurtosis does not depend on time:

The limiting values:

The minimum value of kurtosis:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

## Applications(1)

Generate a sequence of fair coin tosses:

## Properties & Relations(5)

Bernoulli process is weakly stationary:

Bernoulli process has a well-defined StationaryDistribution:

Transition probability does not depend on the current state:

A BinomialProcess is the sum of a BernoulliProcess with :

Accumulate the sample:

Compare to the BinomialProcess:

Align time stamps:

Bernoulli process satisfies the law of large numbers:

The mean function is constant:

Find sample mean:

## Neat Examples(1)

Simulate paths from a Bernoulli process:

Take a slice at 20 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 20:

Wolfram Research (2012), BernoulliProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliProcess.html.

#### Text

Wolfram Research (2012), BernoulliProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliProcess.html.

#### CMS

Wolfram Language. 2012. "BernoulliProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BernoulliProcess.html.

#### APA

Wolfram Language. (2012). BernoulliProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliProcess.html

#### BibTeX

@misc{reference.wolfram_2024_bernoulliprocess, author="Wolfram Research", title="{BernoulliProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BernoulliProcess.html}", note=[Accessed: 14-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_bernoulliprocess, organization={Wolfram Research}, title={BernoulliProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/BernoulliProcess.html}, note=[Accessed: 14-June-2024 ]}