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246 Questions for Chapter 10 Approximation for the Inverse Normal Integral Let Z be a standard normal random variable, that is the probability density function for Z is given by 1 2 1 φ(z) = √ e− 2 z 2π −∞<z <∞ Let Q = P (Z > z), that is Q= Z ∞ z 1 2 1 √ e− 2 z dz 2π The problem is, for a given 0 < Q < 0.5, find the corresponding z that satisfies the above integral equation. For a given 0 < Q < 0.5, the following approximation can be used to compute z : z =t− c0 + c1 t + c2 t2 + ²(Q) 1 + d1 t + d2 t2 + d3 t3 where s 1 Q2 c1 = 0.802853 d1 = 0.189269 t= c0 = 2.515517 d0 = 1.432788 ln c2 = 0.010328 d2 = 0.001308 With this approximation, |²(Q)| < 4.5 × 10−4 . Questions for Chapter 10 10.1 Write a computer program to generate a table for the cumulative distribution function of the standard normal distribution for −3.00 ≤ z ≤ 3.00 in increments of 0.01. 10.2 Let Z ∼ N (0, 1). Write a computer program to find z when P (Z ≤ z) = 0.01, 0.05, 0.25, 0.50, 0.95 and 0.99.