Say I had a question like so:
$X$ ~ $Po(3.1)$
Given $P(X < a) = 0.8$ (1 s.f)
Solve for $a$
I can brute force it by summing the value of the probability mass function at $X=0$, $X=1$, $X=2$ etc. up until I reach the probability of $0.8$
If I do that, then I get: $a = 5$
$$ \sum_{k=0}^{4} [\frac{e^{-3.1} \cdot 3.1^{k}}{k!}] = 0.7981... $$
$ = 0.8$ (1 s.f)
so I get $P(X < 5) = 0.8$ thus $a=5$
My question is: is there a cleaner and non-brute-force method of solving for this?