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From cwinter <chwinterho...@aol.com>
Subject Re: [math] Continuous Distribution
Date Sat, 29 Oct 2011 14:16:15 GMT

Phil Steitz wrote:
> 
> On 10/28/11 9:31 PM, S├ębastien Brisard wrote:
>> Hi,
>> The following question might sound stupid, but occured to me while
>> thinking about MATH-692. So here goes. What was initially meant by
>> "Continuous Distribution" (as in AbstractContinuousDistribution) ?
>> My view on this is that the underlying random variable is defined by a
>> *density*, which takes *continuous* arguments. But nothing prevents
>> this density to be infinite at some *discrete* points (Dirac
>> generalized function). Then the cumulative sum would be only piecewise
>> C1.
>> When these distributions were first implemented, was it intended to
>> include this case?
> 
> We did not talk about these cases initially, but the intent was to
> include all continuous distributions.  More specifically, we did not
> mean to leave a gap - i.e., every distribution should be either
> discrete or continuous, which means singular distributions need to
> be allowed as continuous.
> 
> Phil
> 

Hi,

I also wasn't sure about the interpretation of "continuous" in
ContinuousDistribution for a while. But I was incertain whether the claim
was that the cumulative distribution function should be continuous or the
Distribution itself should be absolutely continuous, i.e. should have a
probability density function. Since density(double) had been put to
ContinuousDistribution I was sure that the scope was absolutely continuous
distributions.

However, when allowing a generalized functions like the delta distribution
(unfortunately, the term "distribution" is overloaded in mathematics) as
density, then any distribution would be a ContinuousDistribution (implying
that there is no need for Distribution, and DiscreteDistribution would be a
special case of ContinuousDistribution). Additionally, it is not possible to
implement such a generalized function meaningfully in the current setting.
Thus I vote for defining ContinuousDistribution to be the interface for
absolutely continuous distribution. I'm fine with a "gap" between
DiscreteDistribution and ContinuousDistribution, i.e. with successors of
Distribution which neither implement DiscreteDistribution nor
ContinuousDistribution.

Best Regards,
Christian


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