I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement.
The formula is applied B times (bootstraped iteration) with replacement in original populations between each iterations.
If I understood well, I can write that inside each iteration, probability for an element to be sampled P(e) is:
And probability of the whole vector S of size n to be sampled P(s) is:
But how to explain that between each iteration B, probabilities P(e) and P(S) are restaured to their origin ?
Is it OK to say that&space;=&space;%5Cfrac%7B(N-n)!%7D%7BN!%7D "P(e) = \frac{(N-n)!}{N!}")
and &space;=&space;%5Cfrac%7B1%7D%7B%5Cbinom%7BN%7D%7Bn%7D%7D "P(S) = \frac{1}{\binom{N}{n}}")
inside an iteration; and that &space;=&space;%5Cfrac%7B1%7D%7BN%7D "P(e) = \frac{1}{N}")
between iteration? And what about P(S) ?