[Rd] pbinom with size argument 0 (PR#8560)
uht@dfu.min.dk
uht at dfu.min.dk
Sun Feb 5 21:40:20 CET 2006
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) =
35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk) FAX: (+45) =
35327907
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