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

__init__(X, P, codomain=None, check=False)

Create the discrete probability space with probabilities on the space X given by the dictionary P with values in the field real_field.

EXAMPLES:

sage: S = [ i for i in range(16) ] 
sage: P = {}
       sage: for i in range(15): P[i] = 2^(-i-1)
sage: P[15] = 2^-16 
sage: X = DiscreteProbabilitySpace(S,P)
sage: X.domain()
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
sage: X.set()
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
sage: X.entropy()
       1.9997253418

A probability space can be defined on any list of elements.

EXAMPLES:

sage: AZ = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'
sage: S = [ AZ[i] for i in range(26) ]
sage: P = { 'A':1/2, 'B':1/4, 'C':1/4 }
sage: X = DiscreteProbabilitySpace(S,P)
sage: X
Discrete probability space defined by {'A': 1/2, 'C': 1/4, 'B': 1/4}
sage: X.entropy()
       1.5
__repr__()
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.

__call__(x)
Return the value of the random variable at x.
__init__(X, f, codomain=None, check=False)

Create free binary string monoid on n generators.

INPUT: x: A probability space f: A dictionary such that X[x] = value for x in X is the discrete function on X

__repr__()
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)

A probability space.

__init__(domain, RR)
A generic probability space on given domain space and codomain ring.
domain()
class sage.probability.random_variable.RandomVariable_generic(X, RR)

A random variable.

__init__(X, RR)
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)

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