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:

$\text{cov}(X,Y) = E((X-E(X)*(Y-E(Y)) = \sum_{x \in S} p(x) (X(x) - E(X))(Y(x) - E(Y))$

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:

$\text{cov}(X,Y) = E((X-E(X)*(Y-E(Y)) = \sum_{x \in S} p(x) (X(x) - E(X))(Y(x) - E(Y))$

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)¶

Random variables and probability spaces¶

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