lect4 - 4 Binomial Random Variable Approximations and...

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1 4. Binomial Random Variable Approximations and Conditional Probability Density Functions Let X represent a Binomial r.v as in (3-42). Then from (2-30) Since the binomial coefficient grows quite rapidly with n , it is difficult to compute (4-1) for large n . In this context, two approximations are extremely useful. 4.1 The Normal Approximation (Demoivre-Laplace Theorem) Suppose with p held fixed. Then for k in the neighborhood of np , we can approximate () = = = = 2 1 2 1 . ) ( 2 1 k k k k n k k k k n q p k n k P k X k P (4-1) ! )! ( ! k k n n k n = n npq PILLAI

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2 (4-2) . 2 1 2 / ) ( 2 npq np k k n k e npq q p k n π Thus if and in (4-1) are within or around the neighborhood of the interval we can approximate the summation in (4-1) by an integration. In that case (4-1) reduces to where We can express (4-3) in terms of the normalized integral that has been tabulated extensively (See Table 4.1). 1 k 2 k ( ) , , npq np npq np + () , 2 1 2 1 2 / 2 / ) ( 2 1 2 2 1 2 2 1 dy e dx e npq k X k P y x x npq np x k k = = π π (4-3) ) ( 2 1 ) ( 0 2 / 2 x erf dy e x erf x y = = π (4-4) . , 2 2 1 1 npq np k x npq np k x = = PILLAI
3 For example, if and are both positive ,we obtain Example 4.1: A fair coin is tossed 5,000 times. Find the probability that the number of heads is between 2,475 to 2,525. Solution: We need Here n is large so that we can use the normal approximation. In this case so that and Since and the approximation is valid for and Thus Here () ). ( ) ( 1 2 2 1 x erf x erf k X k P = 1 x 2 x ). 525 , 2 475 , 2 ( X P (4-5) , 2 1 = p 500 , 2 = np . 35 npq , 465 , 2 = npq np , 535 , 2 = + npq np 475 , 2 1 = k . 525 , 2 2 = k = 2 1 2 . 2 1 2 / 2 1 x x y dy e k X k P π . 7 5 , 7 5 2 2 1 1 = = = = npq np k x npq np k x PILLAI

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4 2 1 ) ( 2 1 ) ( erf 0 2 / 2 = = x G dy e x x y π x erf( x ) x erf( x ) x erf( x ) x erf( x ) 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.01994 0.03983 0.05962 0.07926 0.09871 0.11791 0.13683 0.15542 0.17364 0.19146 0.20884 0.22575 0.24215 0.25804 0.27337 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 0.28814 0.30234 0.31594 0.32894 0.34134 0.35314 0.36433 0.37493 0.38493 0.39435 0.40320 0.41149 0.41924 0.42647 0.43319 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 0.43943 0.44520 0.45053 0.45543 0.45994 0.46407 0.46784 0.47128 0.47441 0.47726 0.47982 0.48214 0.48422 0.48610 0.48778 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 0.48928 0.49061 0.49180 0.49286 0.49379 0.49461 0.49534 0.49597 0.49653 0.49702 0.49744 0.49781 0.49813 0.49841 0.49865 Table 4.1 PILLAI
5 Since from Fig. 4.1(b), the above probability is given by where we have used Table 4.1 4.2. The Poisson Approximation As we have mentioned earlier, for large n , the Gaussian approximation of a binomial r.v is valid only if p is fixed, i.e., only if and what if np is small, or if it does not increase with n ? , 0 1 < x () , 516 . 0 7 5 erf 2 |) (| erf ) ( erf ) ( erf ) ( erf 525 , 2 475 , 2 1 2 1 2 = = + = = x x x x X P 1 >> np . 1 >> npq . 258 . 0 ) 7 . 0 ( erf = Fig. 4.1 x (a) 1 x 2 x 2 / 2 2 1 x e π 0 , 0 2 1 > > x x x (b) 1 x 2 x 2 / 2 2 1 x e π 0 , 0 2 1 > < x x PILLAI

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6 Obviously that is the case if, for example, as such that is a fixed number. Many random phenomena in nature in fact follow this pattern. Total number of calls on a telephone line, claims in an insurance company etc. tend to follow this type of behavior. Consider random arrivals such as telephone calls over a line. Let n represent the total number of calls in the interval From our experience, as we have
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lect4 - 4 Binomial Random Variable Approximations and...

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