Sum and ratio of independent random variables

Sufficient conditions for independence of sum and ratio.

Mathematics
Probability Theory
Author
Published

June 14, 2023

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:

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BibTeX 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/sum-and-ratio-of-independent-random-variables.html},
  langid = {en}
}
For attribution, please cite this work as:
Gherardi, Valerio. 2023. “Sum and Ratio of Independent Random Variables.” June 14, 2023. https://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variables/sum-and-ratio-of-independent-random-variables.html.