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.
The discrete probability space
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
A random variable on a discrete probability space.
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
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:
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
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:
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
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:
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:
A probability space.
A random variable.