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, 

(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.

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