P(x = k | x > 0)

Question

Say that $P$ is a Poisson pmf with parameter $\lambda$. I'm trying to compute $P(x=k, \lambda | x > 0)$ where $k > 0$.

Here's my approach:

$$ P(x=k, \lambda | x > 0) = \frac{P(x = k \,\& \,x > 0, \lambda)}{P(x > 0)}$$

But now I'm stuck on the numerator. How would I compute that?

Answer 1

$P(x=k \land x\gt 0)=P(x=k)$, since $k\gt0$.