# BernoulliProcess

represents a Bernoulli process with event probability p.

# Details # 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:

Introduced in 2012
(9.0)