[ https://issues.apache.org/jira/browse/MATH692?page=com.atlassian.jira.plugin.system.issuetabpanels:commenttabpanel&focusedCommentId=13162487#comment13162487
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Christian Winter commented on MATH692:

As mentioned by Sébastien in MATH699, the implementation of {{IntegerDistribution.inverseCumulativeProbability(double
p)}} can benefit from the ideas which came up for {{RealDistribution.inverseCumulativeProbability(double
p)}} in that thread.
Thus I will remove {{getDomainLowerBound(double p)}} and {{getDomainUpperBound(double p)}}
from the integer distributions. I checked that all current implementations of the lower/upper
bound methods provide the whole support of the distribution as starting bracket. This means
that using {{getSupportLowerBound()}} and {{getSupportUpperBound()}} for the starting bracket
won't degrade the performance of the current distribution implementations. However, a user
might want the improve the performance of his distribution implementations by providing a
more targeted starting bracket for probability {{p}}. Thus I will swap the solving step to
a protected function {{solveInverseCumulativeProbability(double p, int lower, int upper)}},
so that it gets easy to override {{inverseCumulativeProbability}} with an implementation which
finds a better starting bracket.
Furthermore, Phil's idea with Chebyshev's inequality can be applied to the generic implementation
of {{inverseCumulativeProbability}} in order to get a better starting bracket.
> 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
>
> Attachments: Math692_realDomain_patch1.patch
>
>
> 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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