I am considering two discrete random variables, X and Y, with values in {0,1,...,9}.(adsbygoogle = window.adsbygoogle || []).push({});

From my data, I can obtain an estimate for the distribution of Y (the function [itex]P_{Y}(y) = P(Y=y)[/itex] for y in {0,1,...,9}).

I can also obtain experimentally the distribution of the sum [itex]P_{X+Y}(s)[/itex] (that is, P(X+Y = s) for s in {0, 1, ..., 9} (the sum is modulo 10).

From that, I want to solve for distribution function of X ([itex]P_{X}(x) = [/itex] probablility that X = x for x in {0,1,...,9}).

I furthermore assume that X and Y are independent variables, so that their joint probability is the product of their individual probabilities.

From probability theory, I know I can write [itex]P_{X+Y}(s) = \sum_{x = 0}^{9}P_{Y}((s-x) \mod 10) P_{X}(x)[/itex]

Since I want to solve for [itex]P_{X}(x)[/itex], I need to solve a 10x10 system of equations with coefficient matrix given by the values [itex]P_{Y}(s-x)[/itex] and the vector of all values of [itex]P_{X+Y}(s)[/itex] for s in {0,...,9}.

Unfortunately, perhaps due to the estimates not being perfectly accurate, the solution of this system of equations (done numerically) gives me negative values for some of the probabilities in [itex]P_X(x)[/itex].

So is there another way of obtaining an estimate for the values of [itex]P_X(x)[/itex]?

Thank you.

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# Solving for P_x(X) given P_y(Y) and P_(x+y)(X+Y)

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