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 inter-val does not depend on what happened prior to time t. (a) Show that under these conditions f(x, t +t) = f(x, t)[1−α · t]+f(x−1, t)α · t and hence that d[f(x, t)] dt = α[f(x−1, t)−f(x, t)] (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.
Suppose that f(x, t) is the
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