[R] OT: What distribution is this?

[Berwin A Turlach]

G'day Rainer,

On Sat, 25 Sep 2010 16:24:17 +0200
Rainer M Krug <r.m.krug at gmail.com> wrote:

> This is OT, but I need it for my simulation in R.
> 
> I have a special case for sampling with replacement: instead of
> sampling once and replacing it immediately, I sample n times, and
> then replace all n items.
> 
> 
> So:
> 
> N entities
> x samples with replacement
> each sample consists of n sub-samples WITHOUT replacement, which are
> all replaced before the next sample is drawn
> 
> My question is: which distribution can I use to describe how often
> each entity of the N has been sampled?

Surely, unless I am missing something, any given entity would have
(marginally) a binomial distribution:

A sub-sample of size n either contains the entity or it does not.  The
probability that a sub-sample contains the entity is a function of N
and n alone.

x sub-samples are drawn (with replacement), so the number of times that
an entity has been sampled is the number of sub-samples in which it
appears.  This is given by the binomial distribution with parameters x
and p, where p is the probability determined in the previous paragraph.

I guess the fun starts if you try to determine the joint distribution
of two (or more) entities.........

HTH.

Cheers,

	Berwin 

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