Final distributions of $r_1$ and $r_2$ (starting from flat initial distributions of $r_2$ and $\rho$)

For completeness, let us also try to get the closed expressions of $f(r_1\,\vert\,x_1,T_1,x_2,T_2)$ and $f(r_2\,\vert\,x_1,T_1,x_2,T_2)$, although only under the assumption of a flat prior of $\rho$. In this case this choice is forced from the fact that $f_0(\rho)$ cannot be expressed in term of a conjugate prior which would then simplify the calculations. For the general case, in fact, we have to change methods, moving to Markov Chain Monte Carlo (MCMC), as done e.g. in Ref. [1] and as it will be sketched in the next section.

In order to get the pdf of $r_1$, we need to restart from the unnormalized joint distribution (), proceeding then like in Eq. (), but this time integrating over $r_2$ and $\rho$ and absorbing the constant priors in the proportionality factor:

$$f(r_1\,\vert\,x_1,T_1,x_2,T_2) \propto \int_0^\infty\!\!\!\int_0^\infty \tilde f(\ldots) \, \rho\, r_2$$ (108)

$$\propto \int_0^\infty\!\!\!\int_0^\infty\! r_2^{x_2} \cdot e^{-T_2\,r_2} \cdot r_1^{x_1}\! \cdot\! e^{-T_1\,r_1}\cdot \delta(r_1\!-\!\rho\cdot r_2) \, \rho\, r_2$$ (109)

$$\propto \int_0^\infty\!\!\!\int_0^\infty\! r_2^{x_2} \cdot e^{-T_2\,r_2} \cdot r_1^{x_1} \cdot e^{-T_1\,r_1}\cdot \frac{\delta(\rho\!-\!r_1/r_2)}{r_2} \, \rho\, r_2$$ (110)

$$\propto r_1^{x_1} \cdot e^{-T_1\,r_1} \cdot \int_0^\infty\! r_2^{x_2-1} e^{-T_2\,r_2} \, r_2$$ (111)

$$\propto r_1^{x_1} \cdot e^{-T_1\,r_1}\,,$$ (112)

thus reobtaining, besides normalization, Eq. ().

Similarly, we have

$$f(r_2\,\vert\,x_1,T_1,x_2,T_2) \propto \int_0^\infty\!\!\!\int_0^\infty \tilde f(\ldots) \, \rho\, r_1$$ (113)

$$\propto \int_0^\infty\!\!\!\int_0^\infty\! r_2^{x_2} \cdot e^{-T_2\,r_2} \cdot r_1^{x_1}\! \cdot\! e^{-T_1\,r_1}\cdot \delta(r_1\!-\!\rho\cdot r_2) \, \rho\, r_1$$ (114)

$$\propto \int_0^\infty\! r_2^{x_2} \cdot e^{-T_2\,r_2} \cdot (\rho\cdot r_2)^{x_1}\cdot e^{-T_1\,\rho\,r_2}\, \rho$$ (115)

$$\propto r_2^{x_2+x_1} \cdot e^{-T_2\,r_2} \cdot \int_0^\infty\!\! \rho^{x_1}\cdot e^{-T_1\,r_2\,\rho}\, \rho$$ (116)

$$\propto r_2^{x_2+x_1} \cdot e^{-T_2\,r_2} \cdot \frac{\Gamma(x_1+1)}{(r_2\,T_1)^{(x_1+1)}}$$ (117)

$$\propto r_2^{x_2+x_1} \cdot e^{-T_2\,r_2} \cdot r_2^{-(x_1+1)}$$ (118)

$$\propto r_2^{x_2-1} \cdot e^{-T_2\,r_2}\,.$$ (119)

We see that, differently from $f(r_1\,\vert\,x_1,T_1,x_2,T_2)$, the power of $r_2$ is, instead of $x_2$, $x_2\!-\!1$, that is we get an effect similar to that found for the distribution of $\rho$. As a consequence, expected values and standard deviation of $r_2$ are $x_2/T_2$ and $\sqrt{x_2}/T_2$, respectively.