# And none occur in the interval t t δ t the second

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] and none occur in the interval ( t, t t ]. The second way is when t 1 of the arrivals occur in the interval (0 , t ] and one arrival occurs in the interval ( t, t t ]. All other possibilities are ruled out by assumption (b). Assumption (c) implies that the probabilities of these two mutually exclusive or disjoint events are obtained by multiplying the probabilities of the events of the two sub-intervals. The next step in the chain of deductions is to find the derivative of the function f ( x, t ) with respect to t . From equation (7), it follows immediately that (8) df ( x, t ) dt = lim(Δ t 0) f ( x, t + Δ t ) f ( x, t ) Δ t = a f ( x 1 , t ) f ( x, t ) . The final step is to show that the function (9) f ( x, t ) = ( at ) x e at x ! satisfies the condition of equation (8). This is a simple matter of confirming that, according to the product rule of differentiation, we have (10) df ( x, t ) dt = ax ( at ) x 1 e at x ! a ( at ) x e at x ! = a ( at ) x 1 e at ( x 1)! ( at ) x e at x ! = a f ( x 1 , t ) f ( x, t ) . 2

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Exercises 1. The probability density function governing the minutes of time t spent waiting outside a telephone box is given by f ( t ) = ae at .
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• Spring '12
• D.S.G.Pollock
• Poisson Distribution, Probability theory, time interval, ΔT, Binomial Probability Mass

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