Exercise Solution 3.33
Let Z ~ N(0,1) such that:
[s1]
[s2]
From a standard normal table, the .90-quantile of Z is 1.28, so
[s3]
[s4]
[s5]
Accordingly, the .90-quantile Φ−1(.90) of X is 2.38.
Exhibit s1: A standard normal table indicates Φ(1.28) = 0.8997. Rounding to two decimal places, that is Φ(1.28) = 0.90.
A more direct and intuitive solution is to recognize that, as indicated in Exhibit 3.16, the .90-quantile of any normal distribution occurs 1.28 standard deviations above its mean. Accordingly, the .90-quantile of X is
[s6]
Because the handful of quantiles highlighted in Exhibit 16 arise frequently in value-at-risk analyses, its values are worth memorizing.