I have two types of items $X$ and $Y$. I suppose the total number of items, given as the sum of the number of items of type $X$ and number of items of type $Y$, $n_X+n_Y = N$ is fixed, and the number of items of type $X$ ($n_X$) is poisson distributed, namely:
$$P(n_x) = \frac{\exp(-x)x^{n_x}}{n_x!}$$
Is it possible to write the $P(n_y)$ and $P(n_y,n_x)$. I naively thought that since $n_x$ is Poisson distributed with mean $x$ then $P(n_y) = N-P(n_x)$ and $P(n_y,n_x) = P(n_x)(N-P(n_x))$ but clearly these don't normalise to 1.
In fact would it be possible to do this for any arbitrary distribution? How would $E[X^nY^m]$ be affected?
The problem itself does not make much sense. X cannot be Poisson distributed as $X \leq N$ and we know the Poisson distribution can take on any natural number.