Probability of random variable $X$ less than $max(Y_i)$
We have $\Pr[\max(Y_1,Y_2) \leq k] = \Pr[Pois(\lambda) \leq k]^2 = e^{-2\lambda}(\sum_{j=0}^k\frac{\lambda^j}{j!})^2$.

Therefore, $\Pr[X \geq \max(Y_1,Y_2)] = \sum_{k \in \mathbb N} \Pr[X = k] \cdot \Pr[\max(Y_1,Y_2) \leq k] = e^{-(2\lambda+\mu)} \bigg (\sum_{k = 0}^\infty\frac{\mu^k}{k!} \cdot (\sum_{j=0}^k \frac{\lambda^j}{j!})^2 \bigg)$.

From this, if you have some concrete values of $\lambda$ and $\mu$, you can try to see which terms are important, and which are negligible.

I guess the important terms are $k &lt; O(\lambda)$.