With the change of variables-method for a many-to-one transformation function Y = t(X),(adsbygoogle = window.adsbygoogle || []).push({});

what's the logic behind summing the different densities for the roots of x = t^-1(y)?

Probabilities should be ok to add, but densities?

Also, is there no way to extend this method for many-to-many transformation functions, like calculating

the density for each of the roots of y = t(x) separately and summing those sums of densities?

For example, if X has uniform distribution (-1 < x < 1) and Y = +/- sqrt(1-X^2), maybe

it's possible to sum the two densities of positive/negative x for both positive and negative y, then

add the two densities of positive/negative y and divide by two? Would that work if there is asymmetry?

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# I Change of variables many-to-many transformation

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