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May 22nd, 2012, 08:01 PM   #1
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Analytic convariance constant for normal distributions


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...
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June 1st, 2012, 11:12 PM   #2
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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?
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June 4th, 2012, 07:01 PM   #3
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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.
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June 19th, 2012, 04:56 PM   #4
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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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