Let $X$ be Poisson distributed and $\tau \in (0, 1)$. Are there upper bounds of $\min \{n \in \mathbb N : \mathbb P [X \ge n] \le \tau\}$? Let $X$ be a random variable whose distribution is Poisson with parameter $\lambda>0$. Let
$$
A_\tau := \{n \in \mathbb N : \mathbb P [X \ge n] \le \tau\} \quad \forall \tau \in (0, 1).
$$
Are there some upper bounds of $\min A_\tau$ in terms of $\tau$ and $\lambda$? Thank you so much for your elaboration!

A somewhat more complicated upper bound than @Adtiya Dhawan's answer. But the technique is standard, it is the Cramer-Chernoff technique.
We have that
$$ P(X \ge n) \le \inf_{\eta > 0} \exp(-n \eta) E[\exp(\eta X)] = \inf_{\eta > 0} \exp(-n \eta + \lambda(\exp(\eta)-1)) = e^{-\lambda}\left(\frac{e \lambda}{n}\right)^n.$$
Hence, $$\min A_{\tau} \le \inf\left\{n : \left(\frac{e \lambda}{n}\right)^n \le \tau e^{\lambda}\right\}.$$
This would be better than $\lambda / \tau$.
