More on the ratio of Gamma distributed variables

Keeping the notation $\rho_z$ for the ratio of the generic Gamma distributed variables $Z_1$ and $Z_2$, the pdf of Eq. () can be further simplified reminding that the beta special function (or Euler integral of the first kind [33]), defined as

$$(r,s) = \int_0^1 t^{\,r-1} \cdot (1-t)^{s-1}\, t\,,$$ (70)

can be written as

$$(r,s) = \frac{\Gamma(r)\cdot \Gamma(s)}{\Gamma(r+s)}\,.$$ (71)

We can then rewrite the combination of three gamma functions appearing in Eq. () as $1/$B$(\alpha_1,\alpha_2)$, thus getting

$$f(\rho_z\,\vert\,\alpha_1,\beta_1,\alpha_2,\beta_2)\!\! = \frac{1}{\mbox{B}(\alpha_1,\alpha_2)} \cdot \beta_1^{\,\alpha_1}\... ...lpha_1-1}\!\cdot\! (\beta_2+\rho_z\!\cdot\beta_1)^{\,-(\alpha_1+\,\alpha_2)}\,.$$ (72)

As far as mode, expected value and variance are concerned, they can be obtained, without direct calculations, just transforming those of $\rho=r_1/r_2$, seen above, remembering that, starting from a flat prior, $r_i \sim$   Gamma$(\alpha_i=x_i+1, \beta_i=T_i)$. We get then

$$(\rho_z) = \frac{\beta_2}{\beta_1} \cdot \frac{\alpha_1-1}{\alpha_2+1}$$ (73)

$$(\rho_z) = \frac{\beta_2}{\beta_1} \cdot \frac{\alpha_1}{\alpha_2-1} \hspace{5.3cm}(\mathbf{\alpha_2>1})$$ (74)

$$(\rho_z) = \frac{\beta_2^2}{\beta_1^2}\cdot \left[ \frac{\alpha_1}{\alpha_2-... ...\frac{\alpha_1}{\alpha_2-1}\right)\right] \hspace{0.85cm}(\mathbf{\alpha_2>2}).$$ (75)

Moreover, just for completeness, let us mention the special case $\beta_1=\beta_2=1$, that written for the generic variable $X$, becomes

$$f(x\,\vert\,\alpha_1,\alpha_2) \!\! = \frac{1}{\mbox{B}(\alpha_1,\alpha_2)} \cdot \rho_z^{\,\alpha_1-1}\!\cdot\! (1+\rho_z)^{\,-(\alpha_1+\,\alpha_2)},$$ (76)

`known' (certainly not to me before I was attempting to write these subsection) as Beta prime distribution [34], with parameters $\alpha$ and $\beta$:

$$f(x\,\vert\,\alpha,\beta) \!\! = \frac{1}{\mbox{B}(\alpha,\beta)} \cdot x^{\,\alpha-1}\!\cdot\! (1+x)^{\,-(\alpha+\,\beta)}\,.$$ (77)

The name of the distribution is clearly due to the special function resulting from normalization. It is `prime' in order to distinguish it from the more famous (and more important as far as practical applications are concerned) Beta distribution which arises quite `naturally' when inferring the parameter $p$ of the Bernoulli trials, in the light of $x$ successes in $n$ trials (essentially the original problem tackled by Bayes [19] and Laplace [20]), and then used as conjugate prior of the binomial distribution (see e.g. Ref. [30] as well as Ref. [1] for practical applications). The Beta prime distribution is actually what has been independently derived in Sec.  to describe $\rho_\lambda$, although the beta special function was not used there, nor in Sec. .