#9670: Bring probability/random_variable.py to 100% coverage
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Reporter: kcrisman | Owner: mvngu
Type: defect | Status: new
Priority: major | Milestone: sage-5.0
Component: documentation | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Comment(by kohel):
I agree the code (among the first I wrote) should be improved or
rewritten.
Also note that "discrete" should be "finite" -- I never implemented any
code for infinite discrete probability spaces and the finiteness of X a
rather heavy assumption.
That said, despite the lack of documentation, I assert that the above
output is correct. The idea of the discrete or finite probability spaces
is that the underlying set S of X can be anything:
{{{
sage: n = 6 sage: S = list('abcdef')
sage: P = dict([(S[i],1/n) for i in [1..n]])
sage: X = DiscreteProbabilitySpace(S, P)
sage: X.expectation()
0.166666666666667
}}}
It so happens that a probability function IS a random variable so that
one can ask for its expectation. Thus probability space inherits from
random variables which generalize them.
With S so defined, it should be clear that it certainly doesn't make
sense to form \sum_{x \in S} x p(x), since S need not be contained in a
module over the reals for which this sum makes sense. Rather for a
probability space, the expectation is \sum p(x)^2.^
For reasons of avoiding coefficient blowup in the rationals, the default
uses a real ring. This is a design decision which is perhaps questionable,
but avoids problems with undergraduate teaching. If you want the
probability space with value ring in the rationals, then explicitly set
the codomain:
{{{
sage: X = DiscreteProbabilitySpace(S, P, codomain = QQ)
sage: X.expectation()
1/6
}}}
To get the random variable you intended, you need to first create the
probability space as above, then define the random variable with respect
to this:
{{{
sage: f = dict([(S[i],i+1) for i in range(n)])
sage: F = DiscreteRandomVariable(X,f)
sage: F.expectation()
3.50000000000000
}}}
Again, for exact values, set the codomain:
{{{
sage: F = DiscreteRandomVariable(X,f,codomain = QQ)
sage: F.expectation()
7/2
}}}
This two-step creation for a random variable is a bit awkward if all you
want is the uniform probability function. Since this occurs quite often
in practice (for a finite probability space), it might be practical to
have a shortcut which defines the uniform probability space by default.
In that case the syntax might be changed from (X,f,codomain) to
(f,X,codomain), or without change, just letting X be a list or finite set
and setting up the uniform probability on it. This is NOT implemented,
but would be a more intuitive construction for what you intended:
{{{
sage: S = list('abcdef')
sage: F = DiscreteRandomVariable(S,dict([(S[i],i+1) for i in range(n)]))
sage: X = F.probability_space(); X
Discrete probability space defined by {'a': 1/6, 'c': 1/6, 'b': 1/6, 'e':
1/6, 'd': 1/6, 'f': 1/6}
}}}
At the time of writing this (and maybe still true) there was no class of
finite sets and morphisms of sets, which would provide a more natural
input to the constructors for probability space and random variables. It
would be a cleaner construction to have the codomain attached to the
function f or probability function p, and have some general mechanism for
checking that the domains of f and p agree and that codomain of f is a
module over the codomain of p. Python dictionaries were the closest I
could come (without implementing a category of sets) to the datastructure
for morphism of sets.
Currently the domain of a random variable is assumed to be some real ring
or subring but vector-valued random variables would make sense in a
number of settings.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9670#comment:3>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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