Re: Sum and difference of independent variables


Lenore wrote:
Hello everyone,

Is there a proof - or, indeed, does the statement hold at all - that the sum and difference of two independent variables X and Y, (X+Y) and (X-Y), are themselves independent variables, if X and Y are identically distributed?

Thank you.

Lenore

I am not an expert on this subject, but do use random numbers quite
often. I though your question was interesting enough that I wrote a
MathCad *** that generated 1000 uniformly distributed random numbers
X and Y.

I plotted Y vs. X and got what you would expect, 1000 points randomly
spattered on a square. I then plotted Y-X vs Y+X and got 1000 points
randomly spattered on a 45 degree tilted square. It seemed as though
Y-X was independent of Y+X, just like X is independent of Y.

Since the square is tilted at 45 deg, the value of Y-X is limited in
range, depending on the value of Y+X, and vice versa. Maybe this means
they aren't independent, I am just not sure.

I decided to run a correlation. First, X correlated with Y (a product
of the FFT of X with the conjugated of the FFT of Y, transformed back
with an IFFT). The result looked like white noise, i.e., no
correlation. I then did the same with Y-X and Y+X, and again, the
result looked like white noise, though with about twice the RMS
amplitude. I am not sure if that implies lack of independence.

Anyway, it was an interesting problem. I will have to give it more
thought.

.

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