Module: sage.probability.random_variable
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.
Module-level Functions
S) |
X) |
S) |
X) |
Class: DiscreteProbabilitySpace
self, 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.
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.
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
Functions: entropy,
set
self) |
The entropy of the probability space.
self) |
The set of values of the probability space taking possibly nonzero probability (a subset of the domain).
Special Functions: __init__,
__repr__
Class: DiscreteRandomVariable
self, X, f, [codomain=None], [check=False]) |
Create free binary string monoid on
generators.
Input: x: A probability space f: A dictionary such that X[x] = value for x in X is the discrete function on X
Functions: correlation,
covariance,
expectation,
function,
standard_deviation,
translation_correlation,
translation_covariance,
translation_expectation,
translation_standard_deviation,
translation_variance,
variance
self, other) |
The correlation of the probability space X = self with Y = other.
self, other) |
The covariance of the discrete random variable X = self with Y = other.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the variance of
is:
self) |
The expectation of the discrete random variable, namely
,
where
= self and
is the probability space of
.
self) |
The function defining the random variable.
self) |
The standard deviation of the discrete random variable.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the standard deviation of
is defined to be
self, other, map) |
The correlation of the probability space X = self with image of Y = other under map.
self, other, map) |
The covariance of the probability space X = self with image of Y = other under the given map of the probability space.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the variance of
is:
self, map) |
The expectation of the discrete random variable, namely
,
where
= self,
is the probability space of
, and
= map.
self, map) |
The standard deviation of the translated discrete random variable
,
where
= self and
= map.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the standard deviation of
is defined to be
self, map) |
The variance of the discrete random variable
, where
= self,
and
= map.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the variance of
is:
self) |
The variance of the discrete random variable.
Let
be the probability space of
= self, with probability function
,
and
be the expectation of
. Then the variance of
is:
Special Functions: __call__,
__init__,
__repr__
self, x) |
Return the value of the random variable at x.
Class: ProbabilitySpace_generic
self, domain, RR) |
A generic probability space on given domain space and codomain ring.
Functions: domain
Special Functions: __init__
Class: RandomVariable_generic
self, X, RR) |
Functions: codomain,
domain,
field,
probability_space
Special Functions: __init__
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