Re: Std.dev. for missing digits?
- From: "Inf" <infinitysquared@xxxxxxxxx>
- Date: 10 Mar 2007 06:53:07 -0800
On Mar 10, 1:32 pm, Stig Holmquist <stigfjor...@xxxxxxxxxxx> wrote:
The simple formula m(1-1/m)^n tells me how many digits will be missing
on average when sampling digits from a random numbers table if m=10
and n= number of times drawn. This also applies to random drawing of
digits with replacemant in a lottery game.
But how should I calculate the expected std.dev. for any value of n?
Stig Holmquist
Stig,
I would proceed by
1) setting up a random variable representing the number of digits
absent. This is a sum of Kronecker deltas, K= sum_i=1:m (delta(n_i,0))
where n_i is the number of times digit i occurs
2) the number of times the different digits are present follows a
multinomial distribution; each digit has prob 1/m of being present
3) The expectation of K yields m*E[delta(n_1,0)] which yields
the result you give.
4) E[K^2] = m*E[delta(n_1,0)]+m(m-1)E[delta(n_1,0)delta(n_2,0)]
Doing the remaining algebra should allow you to find the variance.
Inf.
.
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