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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.