# Sum and ratio of independent random variables

Sufficient conditions for independence of sum and ratio.

Valerio Gherardi https://vgherard.github.io
2023-06-14

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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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 ...".

### Citation

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/
@misc{gherardi2023sum,
}