[Rd] pbinom with size argument 0 (PR#8560)

[uht@dfu.min.dk]

Hello all

A pragmatic argument for allowing size=3D=3D0 is the situation where the =
size is in itself a random variable (that's how I stumbled over the =
inconsistency, by the way).

For example, in textbooks on probability it is stated that:

  If X is Poisson(lambda), and the conditional=20
  distribution of Y given X is Binomial(X,p), then=20
  Y is Poisson(lambda*p).

(cf eg Pitman's "Probability", p. 400)

Clearly this statement requires Binomial(0,p) to be a well-defined =
distribution.

Such statements would be quite convoluted if we did not define =
Binomial(0,p) as a legal (but degenerate) distribution. The same applies =
to codes where the size parameter may attain the value 0.

Just my 2 cents.

Cheers,

Uffe


-----Oprindelig meddelelse-----
Fra: pd at pubhealth.ku.dk p=E5 vegne af Peter Dalgaard
Sendt: s=F8 05-02-2006 01:33
Til: P Ehlers
Cc: ted.harding at nessie.mcc.ac.uk; Peter Dalgaard; R-bugs at biostat.ku.dk; =
r-devel at stat.math.ethz.ch; Uffe H=F8gsbro Thygesen
Emne: Re: [Rd] pbinom with size argument 0 (PR#8560)
=20
P Ehlers <ehlers at math.ucalgary.ca> writes:

> I prefer a (consistent) NaN. What happens to our notion of a
> Binomial RV as a sequence of Bernoulli RVs if we permit n=3D0?
> I have never seen (nor contemplated, I confess) the definition
> of a Bernoulli RV as anything other than some dichotomous-outcome
> one-trial random experiment.=20

What's the problem ??

An n=3D0 binomial is the sum of an empty set of Bernoulli RV's, and the
sum over an empty set is identically 0.

> Not n trials, where n might equal zero,
> but _one_ trial. I can't see what would be gained by permitting a
> zero-trial experiment. If we assign probability 1 to each outcome,
> we have a problem with the sum of the probabilities.

Consistency is what you gain. E.g.=20

 binom(.,n=3Dn1+n2,p) =3D=3D binom(.,n=3Dn1,p) * binom(.,n=3Dn2,p)

where * denotes convolution. This will also hold for n1=3D0 or n2=3D0 if
the binomial in that case is defined as a one-point distribution at
zero. Same thing as any(logical(0)) etc., really.

--=20
   O__  ---- Peter Dalgaard             =D8ster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) =
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~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) =
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