NormalApprox_Deriv

# NormalApprox_Deriv - Chapter 5 Normal Approximation The...

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Chapter 5 Normal Approximation The normal approximation is a convergent approximation to the binomial probability dis- tribution for arbitrary p as N ! 1 . It was proven for the speci°c case p = 1 2 by DeMoivre in 1733 and for arbitrary p by Laplace in 1812. It is very useful because it eliminates the computational burden of large factorials, which was an even greater issue in the era be- fore modern computing machinery became available. Also, it has fundamental theoretical importance because it leads to the normal density function for continuous random variables. 5.1 Binomial Random Variable For convenience, pertinent results from Chapter 3 are summarized here. By de°nition, a binomial random variable X is the number of successes in N repeated Bernoulli trials; hence X 2 f 0 ; 1 ; 2 ; :::; N g . The binomial probability distribution is P [ X = k ] = ° N k ± p k q N ° k ; k = 0 ; 1 ; :::; N; (5.1) where it has been shown that the mean is ° = Np (5.2) and the variance is ± 2 = Np (1 ° p ) = Npq: (5.3) 5.2 Approximations The normal approximation (known as the DeMoivre-Laplace Theorem) is presented here but proofs are deferred until the end of this chapter. In order to simplify the form of the theorem, the binomial random variable X is °rst standardized: e X : = X ° ° ± . 51

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52 CHAPTER 5. NORMAL APPROXIMATION This standardization has the e/ect of producing the random variable e X which has zero mean and unit variance. Now the speci°c value of e X which corresponds to X = k will be denoted by x k : = k ° ° ± ; k = 0 ; 1 ; :::; N; (5.4) which indicates the number of standard deviations ± by which X deviates from the mean °: Notation 12 The symbol " ± " indicates that the ratio of related quantities approaches one as N ! 1 : Theorem 50 (DeMoivre-Laplace) P [ X = k ] ± " 1 ² p 2 ² ³ ± # e ° ( x k ) 2 2 for j x k j ² A with uniform convergence in the range set by the arbitrary °xed positive constant A: Theorem 51 (Probability by DeMoivre-Laplace Theorem) lim N !1 P h a < e X ² b i = ´ 1 p 2 ² µ Z b a e ° ( x 2 = 2 ) dx; °1 < a < b < 1 : De°nition 38 The normal probability distribution is F X ( x ) : = ´ 1 p 2 ² µ Z x °1 e ° ( u 2 = 2 ) du . De°nition 39 The error function is erf( x ) : = ´ 1 p 2 ² µ Z x 0 e ° ( u 2 = 2 ) du = ) F X ( x ) = 1 2 + erf( x ) for x ³ 0 1 2 ° erf( x ) for x < 0 · where the right-hand relation follows from symmetry of the integrand e ° ( x 2 = 2 ) about x = 0 . Remark 39 The normal probability distribution in De°nition 38 is equivalent to a cumu- lative probability distribution (cdf) for a continuous random variable. With reference to Theorem 51, note that lim N !1 P h a < e X ² b i = F X ( b ) ° F X ( a ) . (5.5) which can be evaluated by using a table of the error function. See Papoulis and Pillai: Table 4-1 (p. 106) where G ( x ) denotes F X ( x ) in De°nition 38.
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