Cumulative probability and inverse cumulative probability inconsistencies

Key: MATH692
URL: https://issues.apache.org/jira/browse/MATH692
Project: Commons Math
Issue Type: Bug
Affects Versions: 2.2, 2.1, 2.0, 1.2, 1.1, 1.0, 1.3, 2.2.1, 3.0
Reporter: Christian Winter
Fix For: 3.0
There are some inconsistencies in the documentation and implementation of functions regarding
cumulative probabilities and inverse cumulative probabilities. More precisely, '<' and
'<=' are not used in a consistent way.
Besides I would move the function inverseCumulativeProbability(double) to the interface Distribution.
A true inverse of the distribution function does neither exist for Distribution nor for ContinuosDistribution.
Thus we need to define the inverse in terms of quantiles anyway, and this can already be done
for Distribution.
On the whole I would declare the (inverse) cumulative probability functions in the basic distribution
interfaces as follows:
Distribution:
 cumulativeProbability(double x): returns P(X <= x)
 cumulativeProbability(double x0, double x1): returns P(x0 < X <= x1) [see also 1)]
 inverseCumulativeProbability(double p):
returns the quantile function inf{x in R  P(X<=x) >= p} [see also 2), 3), and 4)]
1) An aternative definition could be P(x0 <= X <= x1). But this requires to put the
function probability(double x) or another cumulative probability function into the interface
Distribution in order be able to calculate P(x0 <= X <= x1) in AbstractDistribution.
2) This definition is stricter than the definition in ContinuousDistribution, because the
definition there does not specify what to do if there are multiple x satisfying P(X<=x)
= p.
3) A modification could be defined for p=0: Returning sup{x in R  P(X<=x) = 0} would yield
the infimum of the distribution's support instead of a mandatory infinity.
4) This affects issue MATH540. I'd prefere the definition from above for the following reasons:
 This definition simplifies inverse transform sampling (as mentioned in the other issue).
 It is the standard textbook definition for the quantile function.
 For integer distributions it has the advantage that the result doesn't change when switching
to "x in Z", i.e. the result is independent of considering the intergers as sole set or as
part of the reals.
ContinuousDistribution:
nothing to be added regarding (inverse) cumulative probability functions
IntegerDistribution:
 cumulativeProbability(int x): returns P(X <= x)
 cumulativeProbability(int x0, int x1): returns P(x0 < X <= x1) [see also 1) above]

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