# BinomialProcess

represents a binomial process with event probability p.

# Details # Examples

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## Basic Examples(3)

Simulate a binomial process:

Mean and variance functions:

Covariance function:

## Scope(11)

### Basic Uses(5)

Simulate an ensemble of paths:

Compare paths for different values of process parameter:

Process parameter estimation:

Estimate the distribution parameter from sample data:

Correlation function:

Absolute correlation function:

### Process Slice Properties(6)

Univariate SliceDistribution:

Univariate probability density:

Compare to the probability density of BinomialDistribution:

Multi-time slice distribution:

High-order probability density function:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness:

The limiting values:

Find for what values of the parameter BinomialProcess is symmetric:

Kurtosis:

The limiting values:

Find for what values of the parameter BinomialProcess is mesokurtic:

Moment has no closed form for symbolic order:

Generating functions:

CentralMoment has no closed form for symbolic order:

FactorialMoment and its generating function:

Cumulant has no closed form for symbolic order:

## Applications(4)

A quality assurance inspector randomly selects a series of 10 parts from a manufacturing process that is known to produce 20% bad parts. Find the probability that the inspector gets at most one bad part:

The probability of selecting at most one bad part:

It is known that, on average, 50% of the residents in a city like a certain TV program. Find the probability that at least 55% of residents will report that they like a program, in a survey of 804 people from the city:

The probability that at least 55% of the residents in the sample will like the program:

A packet consisting of a string of symbols is transmitted over a noisy channel. Each symbol has a probability 0.0001 of being transmitted in error. Find the largest for which the probability of incorrect transmission (at least one symbol in error) is less than 0.001:

The probability of an error in the transmission:

Plot probability of a transmission error together with error limit:

Find the largest for which the probability of incorrect transmission is less than :

Find the price of a European call option after the third period in a multi-period binomial model, given that the initial price of the underlying is \$100, strike price is \$102, interest rate is 1% per period, and the stock moves up by 7% or down by a factor of 1/1.07:

Model the process:

Price of the call option:

## Properties & Relations(5)

Binomial process is weakly stationary only for p equal to 0:

A BinomialProcess is the sum of a BernoulliProcess with :

Accumulate the sample:

Compare to the binomial process:

Align time stamps:

The time between events in a binomial process follows a PascalDistribution:

Calculate times between changes:

Fit a Pascal distribution:

Compare the data histogram to the estimated probability density function:

Check goodness of fit:

Transition probability between two states:

BinomialProcess is a special case of CompoundRenewalProcess:

The mean functions agree:

Create empirical covariance functions:

Compare the covariance functions:

## Neat Examples(1)

Simulate 500 paths from a binomial process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50: