Imagine a sand clock that drops grains of sand from an upper bulb to the lower one at random intervals.
The probability that $k$ grains of sand are dropped in $t$ seconds is
$$\frac{(\lambda t)^k}{k!}e^{-\lambda t}$$
Assume $\lambda$ is known. How can I estimate the elapsed time when I see $n$ more grains of sand in the lower bulb?
I guess the problem is very common. What is the canonical approach? Should I compute the a confidence interval for $t$? How? Or, what else?
Note that ${{\left( {\lambda t} \right)^{\,k} } \over {k!}}e^{\, - \,\lambda \,t} $ can actually be written as $$ P(k|t) = {{\left( {\lambda t} \right)^{\,k} } \over {k!}}e^{\, - \,\lambda \,t} $$ being the probability of having $k$ grains in a given interval $t$.
$\lambda=1/t_0$ is just a scale factor for $t$, so can take $\tau= \lambda t = t/t_0$ to be a time interval measured in that scale. So $$ P(k|\tau ) = {{\tau ^{\,k} } \over {k!}}e^{\, - \,\,\tau } $$
Suppose $\tau$ to have a uniform a-priori distribution over a (large) interval $T$, then $$ P(k \cap \tau ) = {\tau \over T}{{\tau ^{\,k} } \over {k!}}e^{\, - \,\,\tau } $$ and $$ P(\tau |n) = {{P(n \cap \tau )} \over {P(n)}} = P(\tau \cap n) = {\tau \over T}{{\tau ^{\,n} } \over {n!}}e^{\, - \,\,\tau } $$ will be the probability that, given $n$ grains, the period of time extends till $\tau$ (i.e. the CDF).
Similarly, if the a-priori probability for $\tau$ be a different function.