How would I calculate $P( X \geq x_1 \cap X \geq x_2)$ where $x_1 > x_2$ and $x_1,x_2$ are events and where $P(X = x) = \frac{e^{−\lambda}\lambda^x}{x!}$?
I originally taught the answer would just be $P( X \geq x_1 \cap X \geq x_2) = P( X \geq x_1 )$ but I was reading about the memorylessness property of exponential distributions recently and I'm not sure if it applies to Poissson distributions also.
Clearly, if $x_1,x_2$ are constants, then: $\quad \mathsf P(X\geq x_1\cap X\geq x_2)\\ = \mathsf P(X\geq \min\{x_1,x_2\})\\ = 1-\dfrac{\Gamma(\lfloor\min\{x_1,x_2\}+1\rfloor,\lambda)}{\lfloor\min\{x_1,x_2\}\rfloor!}$
$\Gamma(x,y)$ is the incomplete gamma function