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Unformatted text preview: 1 4. Binomial Random Variable Approximations, Conditional Probability Density Functions and Stirlings Formula Let X represent a Binomial r.v as in (342). Then from (230) Since the binomial coefficient grows quite rapidly with n , it is difficult to compute (41) for large n . In this context, two approximations are extremely useful. 4.1 The Normal Approximation (DemoivreLaplace 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 (41) ! )! ( ! k k n n k n = n npq PILLAI 2 (42) . 2 1 2 / ) ( 2 npq np k k n k e npq q p k n Thus if and in (41) are within or around the neighborhood of the interval we can approximate the summation in (41) by an integration. In that case (41) reduces to where We can express (43) 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 = = (43) ) ( 2 1 ) ( 2 / 2 x erf dy e x erf x y = = (44) . , 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 (45) , 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 . 7 5 , 7 5 2 2 1 1 = = = = npq np k x npq np k x PILLAI 4 2 1 ) ( 2 1 ) ( erf 2 / 2 = = x G dy e x x y 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 4.1where we have used Table 4....
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This note was uploaded on 10/16/2009 for the course EL el6303 taught by Professor Prof during the Spring '09 term at NYU Poly.
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