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: ProbabilitySpace_generic, 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: 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: RandomVariable_generic

A probability space.

domain()#

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

Bases: Parent

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)#