Exercise Solution 3.33

Let Z ~ N(0,1) such that:



From a standard normal table, the .90-quantile of Z is 1.28, so




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


Because the handful of quantiles highlighted in Exhibit 16 arise frequently in value-at-risk analyses, its values are worth memorizing.