Probability of obtaining a greater-than or equal set of observations from a Poisson RV I have a suspicion this might be fairly trivial, but for some reason I cannot obtain a satisfiable answer today.
Assume a Poisson random variable $X$ with known parameter $\lambda$ (though I suspect the result could be extended for any discrete random variable with well-defined CDF)
If one was to sample $n$ values from $X$ to obtain a set $\mathbf{x} = \{x_1, x_2, ..., x_n\}$. What is the probability that $\mathbf{x}$ is greater than or equal to some other set $\mathbf{c} = \{c_1, c_2, ..., c_n\}$, i.e. $\mathbf{x} \ge \mathbf{c}$?
The '$\ge$' operator for sets in this case is defined by comparing the sorted sets elementwise, for instance:
$\{4,3,2\} \geq \{1,2,4\}$ as $[2,3,4] \geq [1,2,4]$ (because $2 \geq 1$, $3 \geq 2$ and $4 \geq 4$),
but
$\{2, 5, 6\} \lt \{3,3,4\}$ since $2 \le 3$ 
 A: I will give some hint towards a solution. I don' t think this is a trivial problem, but some approximate solution should be possible. We have $X_1, \dots, X_n$ iid distributed with Poisson distribution, although I will not use the last part. At the level of generality I will present, we really only need to assume the variables are exchangeable, that is, any permutation of the variables have the same distribution. Denote the probability mass function of $X_1, \dots, X_n$ by $f(x_1, \dots, x_n)$. The order statistics are denoted by $X_{(1)}, \dots, X_{(n)}$.  Then the probability mass function of the order statistics is given by
$$
f(x_{(1)},\dots, x_{(n)}=\begin{cases} n!f(x_{(1)},\dots,x_{(n)}),\text{if}~
   x_{(1)}\le x_{(2)}\le \dots \le x_{(n)} \\
    0, ~\text{otherwise}~ \end{cases}   
$$
Denote the limits by $c_{(1)}\le\dots\le c_{(n)}$ as in the question.  Then, we want the probability 
$$
 P(X_{(1)}\ge c_{(1)}, \dots, X_{(n)}\ge c_{(n)})
$$ and this can be written as a iterated sum ($n$ sum symbols)
$$
\sum_{x_{(n)}=c_{(n)}}^\infty \cdot \sum_{x_{(n-1)}=c_{(n-1)}}^{x_{(n)}} \cdot 
   \dots \cdot \sum_{x_{(1)}=c_{(1)}}^{x_{(2)}} f(x_{(1)},\dots,x_{(n)})
$$
which maybe can be used for small $n$ directly, otherwise you will need to try to find some approximate value, maybe by replacing the sums by integrals? I leave further work for you ...
