I've been looking at detector coincidences and tried to find what general limits apply to coincidences. I was surprised how simply the calculation works out. My question is whether it is correct and where can I find similar stuff ?(adsbygoogle = window.adsbygoogle || []).push({});

Consider two binary sequences produced by random processes where the probabilities of getting 1 are ##p_1## and ##p_2## respectively. Now assume that the number of 1's in the streams is ##1_n \rightarrow Np_n## as ##N \rightarrow \infty##.

If we know the counts ##1_n,\ 0_n=N-1_n## in our sequences then by a permutation argument it is clear that the maximum number of (0,0) coincidences can not be greater than the minimum of ##0_1=N(1-p_1)## and ##0_2=N(1-p_2)##. Similarly the maximum possible (1,1) coincidences is the least of ##Np_1## and ##Np_2##. Assumimg ##p_1<p_2## this gives a total for the (0,0) and (1,1) coincidences of ##S_{12}=N(1-p_2+p_1)##. There are no permutations which give a greater total than this.

The maximum possible correlation between the streams is given by ##\mathcal{C}_{12}=(2S_{12}-N)/N## which gives ##1-2(p_2-p_1)##.

From this one can write for the maximum possible correlations between 4 streams ( assuming ##p_1\leq p_2 \leq p_3 \leq p_4##).

##|\mathcal{C}_{12}+\mathcal{C}_{23}+\mathcal{C}_{34}-\mathcal{C}_{41}| \leq 2##

the ##p_n## terms conveniently cancelling.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Correlation limits for binary variates

**Physics Forums | Science Articles, Homework Help, Discussion**