Ratio of Gamma distributed variables

Having inferred the two rates, we can now evaluate the distribution of $\rho=r_1/r_2$, which is technically just a problem of `direct probabilities', that is getting the pdf $f(\rho\,\vert\,x_1,T_1,x_2,T_2)$ from $f(r_1\,\vert\,x_1,T_1)$ and $f(r_2\,\vert\,x_2,T_2)$ (the Bayesian network that relates the variables of interest is shown in Fig. ).

Figure: Graphical model relating the physical quantities (rates and measurement times) to the observed numbers of events.

We just need to repeat what it has been done in Sec. , taking the advantage of having understood that $f(\lambda_1\,\vert\,x_1)$ and $f(\lambda_2\,\vert\,x_2)$ appearing in Eq. () are indeed Gamma distributions. Therefore, we start evaluating the probability distribution of the ratio of generic Gamma variables, denoted as $Z_1$ and $Z_2$ (and their possible occurrences $z_1$ and $z_2$) in order to avoid confusion with $X$'s, associated so far to measured counts:

$$Z_1\!\! \sim (\alpha_1,\beta_1)$$ (52)

$$Z_2\!\! \sim (\alpha_2,\beta_2)\,.$$ (53)

The pdf of $Z_1/Z_2$ is the given by

$$f(\rho_z\,\vert\,\alpha_1,\beta_1,\alpha_2,\beta_2)\!\! = \int_0^\infty\!\!\!\int_0^\infty\! \delta\!\left(\rho_z-\frac{z_1... ...ot\! f(z_1\,\vert\,\alpha_1,\beta_1)\!\cdot\! f(z_2\,\vert\,\alpha_2,\beta_2)\, z_1 z_2\,,\ \ \ \ \ \$$ (54)

in which we have indicated by $\rho_z$ their ratio. In detail, taking benefit of what we have learned in Sec. ,

$$f(\rho_z\,\vert\,\ldots)\!\! = \int_0^\infty\!\!\!\int_0^\infty\!\!\! z_2\cdot \delta(z_1\!-\!\r... ...,\alpha_2}\cdot z_2^{\,\alpha_2-1}\cdot e^{-\beta_2\,z_2}}{\Gamma(\alpha_2)} \, z_1 z_2$$

$$ = \frac{\beta_1^{\,\alpha_1}\beta_2^{\,\alpha_2}} {\Gamma(\alpha_1)... ...eta_1\,(\rho_z\cdot z_2)}} \cdot {z_2^{\,\alpha_2-1}\cdot e^{-\beta_2\,z_2}} \, z_2$$ (55)

$$ = \frac{\beta_1^{\,\alpha_1}\cdot \beta_2^{\,\alpha_2}} {\Gamma(\al... ...!\! z_2^{\,\alpha_1+\alpha_2-1}\cdot e^{-(\beta_2+\rho_z\,\beta_1)\cdot z_2} \, z_2\,.$$ (56)

Writing $\alpha_1\! +\! \alpha_2$ as $\alpha_*$ and $\beta_2\!+\!\rho_z\!\cdot\! \beta_1$ as $\beta_*$, we get

$$f(\rho_z\,\vert\,\alpha_1,\beta_1,\alpha_2,\beta_2)\!\! = \frac{\beta_1^{\,\alpha_1}\cdot \beta_2^{\,\alpha_2}} {\Gamma(\al... ...,\alpha_1-1} \int_0^\infty\!\!\! z_2^{\alpha_*-1}\cdot e^{-\beta_*\cdot z_2} \, z_2$$ (57)

$$= \frac{\beta_1^{\,\alpha_1}\cdot \beta_2^{\,\alpha_2}} {\Gamma(\al... ...2)}\cdot \rho_z^{\,\alpha_1-1}\cdot \frac{\Gamma(\alpha_*)}{\beta_*^{\alpha_*}}$$ (58)

$$= \frac{\Gamma(\alpha_1+\alpha_2)} {\Gamma(\alpha_1)\cdot \Gamma(\a... ...c{ \rho_z^{\,\alpha_1-1}}{(\beta_2+\rho_z\!\cdot\beta_1)^{\,\alpha_1+\alpha_2}}$$ (59)

$$= \frac{\Gamma(\alpha_1+\alpha_2)} {\Gamma(\alpha_1)\cdot \Gamma(\a... ...dot\! (\beta_2+\rho_z\!\cdot\beta_1)^{\,-(\alpha_1+\,\alpha_2)}\!. \ \ \ \ \ \$$ (60)