# 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)#

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)#

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)#

A probability space.

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

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)#