When Successes and Failures are Independent a Compound Process is Poisson
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.