J. M. STEELE*

Let N be a random variable which takes on nonnegative

integer values, and let X be a random variable f

which takes on values E_1, E_2, ..., E_r.. Now let Y_k

deriote the number of occurrences of event E_k in N

independent trials of the random variable X. If N is

Poisson, it has been observed that f

Y_1, Y_2, . . . , Y_r, are independent. In the case that X is

Bernoulli, and E_1 denotes "success" and E_2 denotes "failure"

this yields the interesting situation that the

variables Y_1 and Y_2=N-Y_1 are independent. This note provides a converse of this fact.

The American Statistician, December 1973, Vol. 27, No. 5 page 232.