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)[source]¶
- Bases: - ProbabilitySpace_generic,- DiscreteRandomVariable- The discrete probability space 
- class sage.probability.random_variable.DiscreteRandomVariable(X, f, codomain=None, check=False)[source]¶
- Bases: - RandomVariable_generic- A random variable on a discrete probability space. - covariance(other)[source]¶
- The covariance of the discrete random variable X = - selfwith 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()[source]¶
- The expectation of the discrete random variable, namely \(\sum_{x \in S} p(x) X[x]\), where \(X\) = - selfand \(S\) is the probability space of \(X\).
 - standard_deviation()[source]¶
- 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)[source]¶
- The correlation of the probability space X = - selfwith image of Y =- otherunder map.
 - translation_covariance(other, map)[source]¶
- The covariance of the probability space X = - selfwith image of Y =- otherunder 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)[source]¶
- 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)[source]¶
- The standard deviation of the translated discrete random variable \(X \circ e\), where \(X\) = - selfand \(e\) = 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)[source]¶
- 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: \[\mathrm{var}(X) = E((X-E(x))^2) = \sum_{x \in S} p(x) (X(x) - E(x))^2\]
 
- class sage.probability.random_variable.ProbabilitySpace_generic(domain, RR)[source]¶
- Bases: - RandomVariable_generic- A probability space.