Sufficient conditions for independence of sum and ratio.
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:
If you see mistakes or want to suggest changes, please create an issue on the source repository.
Text and figures are licensed under Creative Commons Attribution CC BY-SA 4.0. Source code is available at https://github.com/vgherard/vgherard.github.io/, unless otherwise noted. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".
For attribution, please cite this work as
Gherardi (2023, June 14). vgherard: Sum and ratio of independent random variables. Retrieved from https://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variables/
BibTeX citation
@misc{gherardi2023sum, author = {Gherardi, Valerio}, title = {vgherard: Sum and ratio of independent random variables}, url = {https://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variables/}, year = {2023} }