### Navigation

- index
- modules |
- previous |
- Probability »

# 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`()

    x.__init__(...) initializes x; see help(type(x)) for signature

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

  Bases: `sage.structure.parent_base.ParentWithBase`

  A random variable.

  `codomain`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `domain`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `field`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `probability_space`()

    x.__init__(...) initializes x; see help(type(x)) for signature

`sage.probability.random_variable.``is_DiscreteProbabilitySpace`(S)

  x.__init__(...) initializes x; see help(type(x)) for signature

`sage.probability.random_variable.``is_DiscreteRandomVariable`(X)

  x.__init__(...) initializes x; see help(type(x)) for signature

`sage.probability.random_variable.``is_ProbabilitySpace`(S)

  x.__init__(...) initializes x; see help(type(x)) for signature

`sage.probability.random_variable.``is_RandomVariable`(X)

  x.__init__(...) initializes x; see help(type(x)) for signature

### Navigation

- index
- modules |
- previous |
- Probability »