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corr(X,log(1+c exp(X)))
Hi,I am interested in computing the correlation between X and log(1 + c exp(X)) for X being a N(a,b^2) random variable. c is a constant and > 1 (but not very much larger than 1).I thought this would be reasonably simple by expanding the log bit around x=a but I only seem to end up with something like corr(X,const*X)=1.Through numerical experiment with c=1, a=0, b=1 I get a correlation of about 0.955. Increasing c to 1.1 gives a correlation of 0.958.Any ideas??Many thanks!
corr(X,log(1+c exp(X)))
rho = E[(X - a) (Y(X) - ybar)]/(b sig(Y)) = Num/Den = bunch of 1D integrals, immediate in any CAS. (mathematica ,matlab, etc) Num = int (x-a) (Y(x) - ybar) e^{-(x-a)^2/(2 b^2)}/sqrt{2 pi), etc
corr(X,log(1+c exp(X)))
Thanks Alan!I was hoping to find a reasonable analytical approximation, but in principle I think your method would work just as fine in practise (as 1D integrals are quick and easy).Great!