Sum and ratio of independent random variables

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:

• 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: \(\alpha = \beta = 0\).