Random variables and probability spaces

This introduces a class of random variables, with the focus on discrete random variables (i.e. on a discrete probability space). This avoids the problem of defining a measure space and measurable functions.

class sage.probability.random_variable.DiscreteProbabilitySpace(X, P, codomain=None, check=False)

Bases: sage.probability.random_variable.ProbabilitySpace_generic, sage.probability.random_variable.DiscreteRandomVariable

The discrete probability space

entropy()

The entropy of the probability space.

set()

The set of values of the probability space taking possibly nonzero probability (a subset of the domain).

class sage.probability.random_variable.DiscreteRandomVariable(X, f, codomain=None, check=False)

Bases: sage.probability.random_variable.RandomVariable_generic

A random variable on a discrete probability space.

correlation(other)

The correlation of the probability space X = self with Y = other.

covariance(other)

The covariance of the discrete random variable X = self with Y = other.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the variance of \(X\) is:

\[\text{cov}(X,Y) = E((X-E(X)\cdot (Y-E(Y)) = \sum_{x \in S} p(x) (X(x) - E(X))(Y(x) - E(Y))\]

expectation()

The expectation of the discrete random variable, namely \(\sum_{x \in S} p(x) X[x]\), where \(X\) = self and \(S\) is the probability space of \(X\).

function()

The function defining the random variable.

standard_deviation()

The standard deviation of the discrete random variable.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the standard deviation of \(X\) is defined to be

\[\sigma(X) = \sqrt{ \sum_{x \in S} p(x) (X(x) - E(x))^2}\]

translation_correlation(other, map)

The correlation of the probability space X = self with image of Y = other under map.

translation_covariance(other, map)

The covariance of the probability space X = self with image of Y = other under the given map of the probability space.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the variance of \(X\) is:

\[\text{cov}(X,Y) = E((X-E(X)\cdot (Y-E(Y)) = \sum_{x \in S} p(x) (X(x) - E(X))(Y(x) - E(Y))\]

translation_expectation(map)

The expectation of the discrete random variable, namely \(\sum_{x \in S} p(x) X[e(x)]\), where \(X\) = self, \(S\) is the probability space of \(X\), and \(e\) = map.

translation_standard_deviation(map)

The standard deviation of the translated discrete random variable \(X \circ e\), where \(X\) = self and \(e\) = map.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the standard deviation of \(X\) is defined to be

\[\sigma(X) = \sqrt{ \sum_{x \in S} p(x) (X(x) - E(x))^2}\]

translation_variance(map)

The variance of the discrete random variable \(X \circ e\), where \(X\) = self, and \(e\) = map.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the variance of \(X\) is:

\[\mathrm{var}(X) = E((X-E(x))^2) = \sum_{x \in S} p(x) (X(x) - E(x))^2\]

variance()

The variance of the discrete random variable.

Let \(S\) be the probability space of \(X\) = self, with probability function \(p\), and \(E(X)\) be the expectation of \(X\). Then the variance of \(X\) is:

\[\mathrm{var}(X) = E((X-E(x))^2) = \sum_{x \in S} p(x) (X(x) - E(x))^2\]

class sage.probability.random_variable.ProbabilitySpace_generic(domain, RR)

Bases: sage.probability.random_variable.RandomVariable_generic

A probability space.

domain()

class sage.probability.random_variable.RandomVariable_generic(X, RR)

Bases: sage.structure.parent_base.ParentWithBase

A random variable.

codomain()

domain()

field()

probability_space()

sage.probability.random_variable.is_DiscreteProbabilitySpace(S)

sage.probability.random_variable.is_DiscreteRandomVariable(X)

sage.probability.random_variable.is_ProbabilitySpace(S)

sage.probability.random_variable.is_RandomVariable(X)