# 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.

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