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Phil Steitz commented on MATH692:

Have a look at the default implementation of cum(,) now in AbstractDistribution. I think
the incorrectness in the documentation there may have been what triggered Christian to raise
this issue. The equation cum(a,b) = F(b)  F(a) where F is the distribution function is natural
and what the impl there is trying to do. In the discrete case, this equation fails, however,
unless you define the cum to exclude the *lower* endpoint. That's why P(x0 < X <= x1)
is a better definition.
> 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: 1.0, 1.1, 1.2, 1.3, 2.0, 2.1, 2.2, 2.2.1, 3.0
> Reporter: Christian Winter
> Priority: Minor
> 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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