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May 22nd, 2012, 08:01 PM  #1 
Newbie Joined: Apr 2012 Posts: 26 Thanks: 0  Analytic convariance constant for normal distributions
Hi, I've come across a constant that I have found useful in calculations involving normal distributions with sign based peturbations (Cyhelsky skew). However, I haven't been able to calculate it's value analytically, but only experimentally. I was wondering if anyone either knew how to calculate it , or if anyone happens to know a name for it and an analytical expression for it. Given two normal distribution random variables a,b; Each has a mean of 0, a standard deviation of 1, and NO covariance between them; Adding the two variables together will give a result with a standard deviation of (2)**0.5. I am interested in a very similar problem; Take only samples of a and b which have the same sign and add them; reject all samples in a and b which have opposite signs; compute the result. The problem introduces a covariance that is positive, but of an unkown value. I know experimentally, that the result will have a mean of 0, a standard deviation of ~1.809213(2), However, I can't figure out the solution analytically. Is this constant a known value that someone has computed before? If not, can someone give me some pointers on how to calculate the constant? I know how to take the integral of z*exp(0.5*x**2)**2, and then using the area of the bell curve (2*pi)**0.5 to compute the variance or standard deviation of the distribution. I'm just not sure how to set the problem up for two independent normal distributions that are added with the restrictions I have given... 
June 1st, 2012, 11:12 PM  #2 
Newbie Joined: Apr 2012 Posts: 26 Thanks: 0  Re: Analytic convariance constant for normal distributions
O.K. I attempted a solution.... Problem setup, and analogy: I am interested in finding the deviation of adding two normal distributions together that are 100\% correlated in the *Sign* of the data elements, but otherwise are uncorrelated. If I were not adding them together, but just attempting to get the deviation of a normal distribution of single data points; I would approach it as a weighted integral to obtain the average variance, and then convert that to a deviation. The probability of each data element is: and variance is: , which makes the weighted variance: and the total of the weighted variances is therefore: From an integral table... So the final result is 1.0, as expected and Now .......................... ! Trying to do this for an addition, by the same methodology. The relative probability is: But, I only need two quadrants formed by x,y since the data is sign correlated; eg: quadrants I and III. By symmetry, the problem can be reduced to solving only for quadrant I; but keeping in mind that quadrant III makes stay at 0. Computing total area to normalize probability for quadrant I: The variance for addition is: Which makes the weighted variance: So, the combined equation yields for quadrant I: ... and this is as far as I have gotten... Any pointers on solving this last bit? Anyone have Mathematica that can do a brute force? :P 
June 4th, 2012, 07:01 PM  #3 
Newbie Joined: Apr 2012 Posts: 26 Thanks: 0  Re: Analytic convariance constant for normal distributions
I made a mistake about what erf(0) was... and royally messed up the normalization constant of the bell curve, it ought to uniformly be changed to : in the original example. I don't think I made the same mistake in the attempted solution, but if someone sees a mistake  please let me know. 
June 19th, 2012, 04:56 PM  #4 
Newbie Joined: Apr 2012 Posts: 26 Thanks: 0  Re: Analytic convariance constant for normal distributions
I was able to finish solving for it analytically... thanks to some help on another forum. BTW: Does anyone know a name for this constant ? (I'd hate to invent one if there was a standard convention...) Or, perhaps, the Pearson covariance constant equivalent (PPMCC) ? 

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