# Probability Distributions

This section of the user guide covers the probability distribution framework included in the math expressions library.

## Probability Distribution Framework

The probability distribution framework includes many commonly used real and discrete probability distributions, including support for empirical and enumerated distributions that model real world data.

The probability distribution framework also includes a set of functions that use the probability distributions to support probability calculations and sampling.

### Real Distributions

The probability distribution framework has the following functions which support well known real probability distributions:

• `normalDistribution`: Creates a normal distribution function.

• `logNormalDistribution`: Creates a log normal distribution function.

• `gammaDistribution`: Creates a gamma distribution function.

• `betaDistribution`: Creates a beta distribution function.

• `uniformDistribution`: Creates a uniform real distribution function.

• `weibullDistribution`: Creates a Weibull distribution function.

• `triangularDistribution`: Creates a triangular distribution function.

• `constantDistribution`: Creates constant real distribution function.

### Empirical Distribution

The `empiricalDistribution` function creates a real probability distribution from actual data. An empirical distribution can be used interchangeably with any of the theoretical real distributions.

### Discrete

The probability distribution framework has the following functions which support well known discrete probability distributions:

• `poissonDistribution`: Creates a Poisson distribution function.

• `binomialDistribution`: Creates a binomial distribution function.

• `uniformIntegerDistribution`: Creates a uniform integer distribution function.

• `geometricDistribution`: Creates a geometric distribution function.

• `zipFDistribution`: Creates a Zipf distribution function.

### Enumerated Distributions

The `enumeratedDistribution` function creates a discrete distribution function from a data set of discrete values, or from and enumerated list of values and probabilities.

Enumerated distribution functions can be used interchangeably with any of the theoretical discrete distributions.

### Cumulative Probability

The `cumulativeProbability` function can be used with all probability distributions to calculate the cumulative probability of encountering a specific random variable within a specific distribution.

Below is example of calculating the cumulative probability of a random variable within a normal distribution.

``````let(a=normalDistribution(10, 5),
b=cumulativeProbability(a, 12))``````

In this example a normal distribution function is created with a mean of 10 and a standard deviation of 5. Then the cumulative probability of the value 12 is calculated for this specific distribution.

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"b": 0.6554217416103242
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}``````

Below is an example of a cumulative probability calculation using an empirical distribution.

In the example an empirical distribution is created from a random sample taken from the `price_f` field.

The cumulative probability of the value `.75` is then calculated. The `price_f` field in this example was generated using a uniform real distribution between 0 and 1, so the output of the `cumulativeProbability` function is very close to .75.

``````let(a=random(collection1, q="*:*", rows="30000", fl="price_f"),
b=col(a, price_f),
c=empiricalDistribution(b),
d=cumulativeProbability(c, .75))``````

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"b": 0.7554217416103242
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}``````

### Discrete Probability

The `probability` function can be used with any discrete distribution function to compute the probability of a discrete value.

Below is an example which calculates the probability of a discrete value within a Poisson distribution.

In the example a Poisson distribution function is created with a mean of `100`. Then the probability of encountering a sample of the discrete value 101 is calculated for this specific distribution.

``````let(a=poissonDistribution(100),
b=probability(a, 101))``````

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"b": 0.039466333474403106
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}``````

Below is an example of a probability calculation using an enumerated distribution.

In the example an enumerated distribution is created from a random sample taken from the `day_i` field, which was created using a uniform integer distribution between 0 and 30.

The probability of the discrete value 10 is then calculated.

``````let(a=random(collection1, q="*:*", rows="30000", fl="day_i"),
b=col(a, day_i),
c=enumeratedDistribution(b),
d=probability(c, 10))``````

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"d": 0.03356666666666666
},
{
"EOF": true,
"RESPONSE_TIME": 488
}
]
}
}``````

### Sampling

All probability distributions support sampling. The `sample` function returns 1 or more random samples from a probability distribution.

Below is an example drawing a single sample from a normal distribution.

``````let(a=normalDistribution(10, 5),
b=sample(a))``````

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"b": 11.24578055004963
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}``````

Below is an example drawing 10 samples from a normal distribution.

``````let(a=normalDistribution(10, 5),
b=sample(a, 10))``````

When this expression is sent to the `/stream` handler it responds with:

``````{
"result-set": {
"docs": [
{
"b": [
10.18444709339441,
9.466947971749377,
1.2420697166234458,
11.074501226984806,
7.659629052136225,
0.4440887839190708,
13.710925254778786,
2.089566359480239,
0.7907293097654424,
2.8184587681006734
]
},
{
"EOF": true,
"RESPONSE_TIME": 3
}
]
}
}``````

### Multivariate Normal Distribution

The multivariate normal distribution is a generalization of the univariate normal distribution to higher dimensions.

The multivariate normal distribution models two or more random variables that are normally distributed. The relationship between the variables is defined by a covariance matrix.

#### Sampling

The `sample` function can be used to draw samples from a multivariate normal distribution in much the same way as a univariate normal distribution.

The difference is that each sample will be an array containing a sample drawn from each of the underlying normal distributions. If multiple samples are drawn, the `sample` function returns a matrix with a sample in each row. Over the long term the columns of the sample matrix will conform to the covariance matrix used to parametrize the multivariate normal distribution.

The example below demonstrates how to initialize and draw samples from a multivariate normal distribution.

In this example 5000 random samples are selected from a collection of log records. Each sample contains the fields `filesize_d` and `response_d`. The values of both fields conform to a normal distribution.

Both fields are then vectorized. The `filesize_d` vector is stored in variable `b` and the `response_d` variable is stored in variable `c`.

An array is created that contains the means of the two vectorized fields.

Then both vectors are added to a matrix which is transposed. This creates an observation matrix where each row contains one observation of `filesize_d` and `response_d`. A covariance matrix is then created from the columns of the observation matrix with the `cov` function. The covariance matrix describes the covariance between `filesize_d` and `response_d`.

The `multivariateNormalDistribution` function is then called with the array of means for the two fields and the covariance matrix. The model for the multivariate normal distribution is assigned to variable `g`.

Finally five samples are drawn from the multivariate normal distribution.

``````let(a=random(collection2, q="*:*", rows="5000", fl="filesize_d, response_d"),
b=col(a, filesize_d),
c=col(a, response_d),
d=array(mean(b), mean(c)),
e=transpose(matrix(b, c)),
f=cov(e),
g=multiVariateNormalDistribution(d, f),
h=sample(g, 5))``````

The samples are returned as a matrix, with each row representing one sample. There are two columns in the matrix. The first column contains samples for `filesize_d` and the second column contains samples for `response_d`. Over the long term the covariance between the columns will conform to the covariance matrix used to instantiate the multivariate normal distribution.

``````{
"result-set": {
"docs": [
{
"h": [
[
41974.85669321393,
779.4097049705296
],
[
42869.19876441414,
834.2599296790783
],
[
38556.30444839889,
720.3683470060988
],
[
37689.31290928216,
686.5549428100018
],
[
40564.74398214547,
769.9328090774
]
]
},
{
"EOF": true,
"RESPONSE_TIME": 162
}
]
}
}``````