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240 Chapter 4 Special Distributions Proof One of the ways to verify Theorem 4.3.1 is to show that the limit of the 2 2 X â np moment-generating function for ânp(1 as n â â is et /2 and that et /2 is also â p) &â 2 the value of ââ et z Â· â12Ï eâz /2 dz. By Theorem 3.12.2, then, the limiting pdf of X ânp Z = ânp(1â is the function f Z (z) = â12Ï eâz p) for the proof of a more general result. 2 /2 , ââ < z < â. See Appendix 4.A.2 Comment We saw in Section 4.2 that Poissonâs limit is actually a special case of âÎ» k Poissonâs distribution, p X (k) = e k!Î» , k = 0, 1, 2, . . . . Similarly, the DeMoivre-Laplace limit is a pdf in its own right. Justifying that assertion, of course, requires proving 2 that f Z (z) = â12Ï eâz /2 integrates to 1 for ââ < z < â. Curiously, there is no algebraic or trigonometric substitution that can be used to demonstrate that the area under f Z (z) is 1. However, by using polar coordinates, we can verify a necessary and sufï¬cient alternativeânamely, that the square of & â 1 âz 2 /2 â dz equals 1. ââ 2Ï e To begin, note that % â % â % â% â 1 1 1 1 2 2 2 2 eâx /2 d x Â· â eây /2 dy = eâ 2 (x +y ) d x d y â 2Ï ââ ââ 2Ï ââ 2Ï ââ Let x = r cos Î¸ and y = r sin Î¸ , so d x d y = r dr dÎ¸ . Then % â% â % 2Ï % â 1 1 1 2 2 2 eâ 2 (x +y ) d x d y = eâr /2 r dr dÎ¸ 2Ï ââ ââ 2Ï 0 0 % 2Ï % â 1 2 r eâr /2 dr Â· dÎ¸ = 2Ï 0 0 =1 Comment The function f Z (z) = â12Ï eâz 2 /2 is referred to as the standard normal (or Gaussian) curve. By convention, any random variable whose probabilistic behavior is described by a standard normal curve is denoted by Z (rather than X , Y , or W ). 2 Since M Z (t) = et /2 , it follows readily that E(Z ) = 0 and Var(Z ) = 1. Finding Areas Under the Standard Normal Curve In order to use Theorem 4.3.1, we need to be able to ï¬nd the area under the graph of f Z (z) above an arbitrary interval [a, b]. In practice, such values are obtained in one of two waysâeither by using a normal table, a copy of which appears at the back of every statistics book, or by running a computer software package. Typically, both approaches give the cdf, FZ (z) = P(Z â¤ z), associated with Z (and from the cdf we can deduce the desired area). Table 4.3.1 shows a portion of the normal table that appears in Appendix A.1. Each row under the Z heading represents a number along the horizontal axis of f Z (z) rounded off to the nearest tenth; Columns 0 through 9 allow that number to be written to the hundredths place. Entries in the body of the table are areas under the graph of f Z (z) to the left of the number indicated by the entryâs row and column. For example, the number listed at the intersection of the â1.1â row and the â4â column is 0.8729, which means that the area under f Z (z) from ââ to 1.14 is 0.8729. That is, % 1.14 1 2 â eâz /2 dz = 0.8729 = P(ââ < Z â¤ 1.14) = FZ (1.14) 2Ï ââ