On a characterization of the Poisson distribution and its statistical applications

The distributions of $n$ mutually independent random variables $X_1,\dots,X_n$ are Poisson ones if and only if the conditional joint distribution of $X_1,\dots,X_n$ given $\Sigma X_i=K$ is the multinomial distribution (4). If we wish to test the hypothesis that $X_1,\dots,X_n$ are Poisson random variables we can use the conditional test (8). This test considered as an unconditional one is asymptotically the most powerful test against close binomial or negative binomial alternatives. The characterization of the Poisson distribution and its extensions for the binomial and the negative binomial distributions can be used to generate Poisson, binomial or negative binomial random numbers.