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As a follow up to algebra-of-random-variables, is it possible to compute:

$z = \frac{dx}{dy}$

Where $x$ and $y$ are drawn from a known, independent probability distribution $x \in f(x)$, $y \in g(y)$? Geometrically the problem makes sense, one has a function $h(x,y) = f(x)g(y)$ that is the joint probability distribution and we are essentially asking for the gradient along one direction. I can get this to the expression ($z \in q(z)$):

$q(z) = \int f(x) g( \int{[z dx]} ) ( \frac{d}{dz} \int{[z dx]} ) dx$

But I'm not quite sure what to make of this, or even how to compute it given specific PDF's.

EDIT: For simplicity, assume all the PDF's considered are continuous and smooth.

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up vote 4 down vote accepted

Radon-Nikodym derivative of the measures represented by variables $x$ and $y$ is one answer to your question, and is somewhat similar to the reasoning you sketched about a product density $h(x,y)=f(x)g(y)$. For "change in $x$ per change in $y$" you perhaps want the correlation coefficient (multiplied by standard deviation of $x$).

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You can take a derivative of a function $x$ with respect to one of its variables and the function has to be smooth w.r.t. this parameter. If $x$ is being drawn from a set of smooth functions then it is certainly possible to consider the derivative of a random variable $x$.

However, I'm really not sure what $\frac{dx}{dy}$ would even mean in your case.

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