Inferring $r$, having observed $x$ counts in the measuring time $T$

Being $r$ equal to $\lambda/T$, we can obtain its pdf by a simple change of variables.¹⁸But, having practiced a bit with the Gamma distribution, we can reach the identical result observing that, using again a flat prior and neglecting irrelevant factors, the pdf of $r$ is given by

$$f(r\,\vert\,x,T) \propto f(x\,\vert\,r,T)$$ (28)

$$\propto (r\cdot T)^x\cdot e^{-r\,T}$$ (29)

$$\propto r^x\cdot e^{-T\,r},$$ (30)

in which we recognize, besides the normalization factor, a Gamma pdf for the variable $r$ with $\alpha=x+1$ and $\beta=T$, and hence

$$f(r\,\vert\,x,T) = \frac{\beta^{\,\alpha}\cdot r^{\,\alpha-1}\cdot e^{-\beta\,r}}{\Gamma(\alpha)}$$ (31)

$$= \frac{T^{\,x+1}\cdot r^{\,x}\cdot e^{-T\,r}}{x!}\,.$$ (32)

Mode, expected value and standard deviation of $r$ are then (see Appendix A)

$$(r) = \frac{\alpha-1}{\beta} = \frac{x}{T}$$

$$(r) = \frac{\alpha}{\beta} = \frac{x+1}{T}$$

$$\sigma(r) = \frac{\sqrt{\alpha}}{\beta} = \frac{\sqrt{x+1}}{T}\,,$$

as also expected from the `summaries' of $f(\lambda\,\vert\,x)$ and making use of $r=\lambda/T$.
$[$ Note that the pdf () assumes, as explicitly written in the condition, a precise value of $T$. If this is not the case and $T$ is uncertain, then, similarly to what we have seen in footnote , the pdf of $r$ is evaluated as $f(r\,\vert\,x,I) = \int_{0}^\infty f(r\,\vert\,x,T,I)\cdot f(T\,\vert\,I)\,$d$T$. $]$