# Thread: Runs test (Wald-Wolfowitz) distribution

1. ## Runs test (Wald-Wolfowitz) distribution

Hi, I have a problem finding the proof of 2 formulas:
I'm studying the Wald-Wolfowitz Runs test (often known simply as "run test"), description is on this page:
Wald-Wolfowitz runs test - Wikipedia, the free encyclopedia

The formulas for the mean and the variance of the distribution of the number of runs are given without proof. Where can I find it?

Thanks!

2. Originally Posted by rargh
Hi, I have a problem finding the proof of 2 formulas:
I'm studying the Wald-Wolfowitz Runs test (often known simply as "run test"), description is on this page:
Wald-Wolfowitz runs test - Wikipedia, the free encyclopedia

The formulas for the mean and the variance of the distribution of the number of runs are given without proof. Where can I find it?

Thanks!
Feller gives the PMF for the number of runs, but not the mean and variance, if that helps. Let M be the number of +s and N be the number of -s. Let (M)C(N) = M!/N!(M-N)!. Then for integer v,
P(2v) = 2.(M-1)C(v-1).(N-1)C(v-1)/(M+N)C(M).
P(2v+1) = {(M-1)C(v).(N-1)C(v-1)+(M-1)C(v-1).(N-1)C(v)}/(M+N)C(M).

3. Thanks, you really saved me... and such a quick answer!

The PMF is even better!

The solution was not too difficult to understand, but still I could have never managed to find it on my own... believe me I was trying to do it but I was waaaaay off the right reasoning.

This forum might really save me, I've got this statistical quality control exam with dozens of unproved formulas. I hope I will make it through...

Thanks for everything!

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# wald-wolfowitz test proof

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