Suppose that f(x, t) is the probability of getting x successes during a time interval of length t when (i) the probability of a success during a very small time interval from t to t + ?t is α · ?t, (ii) the probability of more than one success during such a time interval is negligible, and (iii) the probability of a success during such a time interval does not depend on what happened prior to time t.

(a) Show that under these conditions

and hence that

(b) Show by direct substitution that a solution of this infinite system of differential equations (there is one for each value of x) is given by the Poisson distribution with λ = αt.