Your help in this problem is highly appreciated.
Suppose we are given a multinomial probability distribution
N!
P(X_1=N_1, X2=N_2,..., X_K=N_K) = ---------------- p^N_1*q^N_2*q^N_3...*q^N_K,
N_1! N_2!...N_K!
where p + q*(K-1)=1, and N_1 + N_2+ N_K = N.
How to compute an approximation of the following
probability, where {X_i}s are distributed as above:
P( X_1 = max{X_1, X_2,..., X_K} )?
if p>q or if p > q*(K-1)
For example, when K=2 we simply have (assume N is even):
N N!
P( X_1 = max{X_1, X_2} ) = P( X_1 >= X_2 ) = Sum -------- p^i*(1-p)^(N-i)
i=N/2 i!(N-i)!
Are any approximations known for such sums?
What about the case of general K?
In my applications N is on order of several hundreds,
so exact computation is not feasible.
Dan
.
.
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