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# stark_and_woods_approx - Sec 1.10 ASYMPTOTIC BEHAVIOR OF...

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1.10 ASYMPTOTIC BEHAVIOR OF THE BINOMIAL LAW: THE POISSON LAW Suppose that in the binomial function b(k; n,p),n»l,p<<l, but np remains constant, say np = a. Recall that q = 1 — p. Hence (I)Ai-p)-^^(i-J) n—k

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40 Chapter 1 Introduction to Probability where n(n - 1)... (n - k + 1) ^ nk if n is allowed to become large enough and k is held fixed. Hence in the limit as n -> oo, p -+ 0, and k « n, we obtain n — k n ^ Thus in situations where the binomial law applies with n » 1, p « 1 but np = a is a finite constant, we can use the approximation b ( k ; n l P ) ^ ^ r e - a . ( 1 - 1 0 - 2 ) fc! E x a m p l e 1 . 1 0 - 1 . " ~ ~ A computer contains 10,000 components. Each component fails independently from the others and the yearly failure probability per component is 10 4. What is the probability that the computer will be working one year after turn-on? Assume that the computer fails if one or more components fail. Solution p = 10-4 n = 10,000, k = 0, np = 1. H e n c e 0 - 6(0; 10,000,10"4) = Qj-e"l- = °-368- E x a m p l e 1 . 1 0 - 2 — " — ' Suppose that n independent points are placed at random in an interval (0, T). Let 0 < h < t2<T and t2 - h = r. Let r/T « 1 and n » 1. What is the probability of observing exactly k points in r seconds? (Figure 1.10-1.) Solution Consider a single point placed at random in (0, T). The probability of the point appearing in r is r/T. Let p = r/T. Every other point has the same probability of being in r seconds. Hence, the probability of finding k points in r seconds is the binomial law P [ k p o i n t s i n r s e c ] = ( f \ p k q n ~ k . ( 1 - 1 0 - 3 ) With n » 1, we use the approximation in Equation 1.10-1 to give 6 ( f c ; n , p ) . ( ^ ) f c ^ - , ( 1 - 1 0 - 4 )
Sec. 1.10. ASYMPTOTIC BEHAVIOR OF THE BINOMIAL LAW: THE POISSON LAW 41 where n/T can be interpreted as the "average" number of points per unit interval. Equations 1.10-1 and 1.10-4 are examples of the Poisson probability law. The Poisson law with parameter a (a > 0) is defined by* fc P [ k p o i n t s ] = e ~ a % r ( 1 . 1 0 - 5 ) k\ where fc = 0,1,2, With a = Ar, where A is the average number of events per unit time and r is the length of the interval (£, t + r), the probability of A; events in r is P ( f c ; M + r ) = e " A r ^ - . ( 1 . 1 0 - 6 ) k\ In Equation 1.10-6 we assumed that A was independent of t. If A depends on t, the product Ar gets replaced by the integral ff+T A(£) d£, and the probability of k events in the interval (t, t + r) is P(fc; t,t + r) = exp / t + T " I - I T r t + T - I k A t f K £ i y a ^ (1.10-7) The Poisson law P[k events in Ax] or more generally P[k events in (x, x + Ax)] where x is time, volume, distance, and so forth, and Ax is the interval associated with x is widely used in engineering and sciences. Some typical situations in various fields where the Poisson law is applied are listed below. Physics. In radioactive decay—P[k alpha particles in r seconds] with A the average number of emitted alpha particle per second. Operations Research. In planning the size of a switchboard—P[k telephone calls in r seconds] with A the average number of calls per second.

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stark_and_woods_approx - Sec 1.10 ASYMPTOTIC BEHAVIOR OF...

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