TITLE: The probability of maximum of two independent Poisson-distributed variables
QUESTION [3 upvotes]: I have two Poisson distributions with parameters $\lambda_1$ and $\lambda_2$ and two independent variables $A$ and $B$ from these two distributions. I know that 
$$\begin{equation*}
\mathsf P(\min(A,B)>K)=\mathsf P(A>K) \ast \mathsf P(B>K)  
\end{equation*}$$
I'm just wondering what $\mathsf P(\max(A,B)>K)$ is. Thanks.

REPLY [4 votes]: If the maximum of two numbers is greater than some constant, then either one or the other of the numbers is greater than it;   they are not both less than or equal to it.
If the random variables are independently Poisson distributed then:
$$\begin{align}
\mathsf P(\max(A,B)>K) & =\mathsf P(A>K \cup B>K)
\\[1ex] & = 1 - \mathsf P(A\leq K)\mathsf P(B\leq K)
\\[1ex] & = 1 - \sum_{j=0}^K\frac{\lambda_1^j e^{-\lambda_1}}{j!}\;\sum_{j=0}^K\frac{\lambda_2^j e^{-\lambda_2}}{j!}
\\[1ex] & = 1 - \frac{\Gamma(K+1,\lambda_1)\Gamma(K+1,\lambda_2)}{K!^2}
\end{align}$$

$\Gamma(\cdot,\cdot)$ is the Incomplete Gamma Function
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