Sum and ratio of independent random variables

Mathematics Probability Theory

Sufficient conditions for independence of sum and ratio.

Valerio Gherardi

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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For attribution, please cite this work as

Gherardi (2023, June 14). vgherard: Sum and ratio of independent random variables. Retrieved from

BibTeX citation

  author = {Gherardi, Valerio},
  title = {vgherard: Sum and ratio of independent random variables},
  url = {},
  year = {2023}