Let \(X\) and \(Y\) be two continuous independent random variables, with joint density \(f_{XY}(x,y)=f_X(x)f_Y(y)\). Define: \[ s = x+y, \qquad r = x/y, \] with inverse transformation given by:
\[ y = \frac{s}{1+r},\qquad x = \frac{rs}{1+r}. \] The Jacobian of the \((x,y) \mapsto (s,r)\) transformation is:
\[ \left|\dfrac{\partial (s,r)}{\partial(x,y)}\right|= \dfrac{(1+r)^2}{s}. \] Hence the joint density of \(S = X+Y\) and \(R = X/Y\) is given by:
\[ f_{SR}(s,r) = f(x,y)\left|\dfrac{\partial (x,y)}{\partial(s,r)}\right|=f_X(\frac{rs}{1+r})f_Y(\frac{s}{1+r})\frac{s}{(1+r)^2}. \]
The necessary and sufficient condition for this to factorize into a product, \(f_{SR}(s,r)\equiv f_S(s)f_R(r)\), is that \(f_X(x)f_Y(y) = g_S(s)g_R(r)\) for some functions \(g_S\) and \(g_R\).
This is true for all functions \(f_X\) and \(f_Y\) from the family:
\[ \phi(t) = \text{const} \times t^\alpha e^{-\beta t}. \] This includes some important special cases:
- The \(\chi ^2\) distribution (\(\alpha = \frac{\nu}{2}-1,\,\beta = \frac{1}{2}\)).
- The exponential distribution: \(\alpha = 0,\,\beta >0\).
- The “homogeneous” distribution: \(\beta = 0\) (restricted to the appropriate domain).
- The uniform distribution: \(\alpha = \beta = 0\).
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Citation
@online{gherardi2023,
author = {Gherardi, Valerio},
title = {Sum and Ratio of Independent Random Variables},
date = {2023-06-14},
url = {https://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variables/},
langid = {en}
}