I am currently revisiting stats since I'm really rusty on the topic. When I came across the formula for the density of the product distribution
$$ f_Z(z) = \int\limits_{-\infty}^{+\infty}f_X(x)f_Y(\tfrac zx)\frac{1}{|x|}dx $$
Here something bugged me: although the premise $Z=XY$ is symmetric w.r.t. permutation of the variables $X,Y$, the final formula is not. So I wondered whether there is a formula that does; Surely enough, we can write the cdf as:
$$ F_Z(z) = \int\limits_{-\infty}^{+\infty} \int\limits_{-\infty}^{+\infty} f_X(x)f_Y(y)\theta(z-xy) dx dy $$
where $\theta$ is the heaviside step function. Here, seeing the $z-xy$ term I wondered whether any simplifications could be made using hyperbolic coordinates. If we set
$$ \begin{matrix} x &=& \sqrt{w} e^t\\ y &=& \sqrt{w}e^{-t}\end{matrix} $$
we find that the functional determinant to be
$$ \left| \begin{matrix} \frac{1}{2\sqrt w} e^t & \sqrt{w}e^{t} \\ \frac{1}{2\sqrt w}e^{-t} & \sqrt{w} e^{-t}\end{matrix} \right| = -1$$
and hence:
$$ F_Z(z) = \int\int f_X(\sqrt{w} e^{t})f_Y(\sqrt{w} e^{-t})\theta(z-w)dw dt $$
so the density, given by differentiation wrt $z$, is
\begin{align} f_Z(z) &= \int\Big[\frac{d}{dz}\int_{-\infty}^z f_X(\sqrt{w} e^{t})f_Y(\sqrt{w} e^{-t}) dw\Big] dt \\ &= \int\limits_{-\infty}^{+\infty} f_X(\sqrt{z} e^{t})f_Y(\sqrt{z} e^{-t}) dt \end{align}
Which is a formula that respects the symmetry! (modulo time reversal $t\to - t$)
Sanity check: Let $X,Y$ be iid uniform on $[0,1]$, then $0\le \sqrt{z}e^t \le 1 \iff t \le -\frac 12 \log(z)$ and $0\le \sqrt z e^{-t} \le 1 \iff t\ge \frac 12 \log z$. Also, by taking the product of inequalities we have $0\le z\le 1$. Combining these we have
\begin{align} f_Z(z) &= \int_{-\infty}^{+\infty} 1_{[0,1]}(\sqrt{z} e^{t}) 1_{[0,1]}(\sqrt{z} e^{-t}) dt \\ &= \int_{+\frac 12 \log z}^{-\frac 12 \log z} 1_{[0,1]}(z) dt \\ &= - \log z \cdot 1_{[0,1]}(z) \end{align}
which is indeed the correct result (cf. this thread)
Questions:
Is my calculation correct and rigorous? (I recon there may be a problem with the sign of $w$ and $z$, so it may only work for non-negative values).
Is this equation known? Are there known extensions to the case with $N$ variables?
Finally I am looking for a book recommendation where they in detail deal with product and ratio distributions.
Thanks for reading and any comments & thoughts!
