Let me answer the question in the case n=2, n'=1. If I use the random permutation strategy we've discussed (to compute Y by making use of X and some extra randomness), then P(Y=1) = P(X=2)*1 + P(X=1)*1/2 + P(X=0)*0 = p^2 + 2p(1-p)/2 + 0 = p^2 + p-p^2 = p, so Y is governed by the Binomial(p,1) distribution. Jim On 6/24/12, Guy Haworth <g.haworth@reading.ac.uk> wrote:
Maybe I've misunderstood the question, but I'm wondering if JP is asking for something that cannot exist.
Suppose a 'coin', simulated for example by a random-number generator on a computer, has a probability 'p' of coming up Heads.
Then the probability of getting 'X' Heads out of 'n' is defined by a binomial formula involving 'p', 'X' and 'n'.
However, if I observe 'X' heads occurring in 'n' tosses of the coin, I have a different piece of information - in fact less information.
Even if X/n = p, I don't know that p is the probability of coming up Heads. So, in terms of knowledge, probability and statistics, I'm 'in a different place'.
The case of getting HH on two coin tosses (with probability p^2) shows this. The case of getting H on one coin toss shows this too.
If I could derive a binomial formula for the number of Heads in n' tosses of the coin, it would be equivalent to knowing what the probability of Heads is on each toss.
But I don't know this, so I can't derive a binomial formula from observing a sequence of coin-tosses.
Guy
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