I have the following query related to probability distributions which arise
from sampling.
I imagine that I have a population of N elements and I have r successes.
Firstly I disribute the r successes randomly among the N elements in two
fashion - with replacement and without replacement. If I distribute them
without replacemnt then I randomly pick an element and attribute the
proporty of success to it. I do not replace replace this element and I
select another etc. When I allocate the successes without replacement then
each element has either 1 success or 0 successes attributed to it. With
replacement, an element can have more than 1 success attributed to it. Let
me denote these two schemes as pop with rep and pop without rep.
Now I imagine that I take a sample n from the population and count the
number of successes. I can take the sample with replacement or without
replacement. I am interested in the probability distribution of the number
of successes.
With pop without rep and a sample taken with replacement the probability
distribution is just Binomial (n,p=1/r)
With pop without rep and a sample taken without replacement the probability
distribution is hypergeometric.
With pop with rep (multiple successes possible per element) and sampling
with replacement, I find that the distribution of successes for a
particular element is binomial and the problem becomes a messy one of
finding all the possible combinations. For instance a total number of
successes of 3 could be made up of three seperate elements each with 1
success or 2 elements one with 2 successes the other with 1 or one element
with 3 successes.
With pop with rep and sampling without replacement I am stuck. Can anyone
point me in the right direction. I feel that the last two examples I have
described must be quite common in applications and I would welcome any
advice on possible approximations or references.
Thanks
Glenn Treacy
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