Exercise Solution 3.40

1. By [3.128], μ = ξ′μ = 0.4.
2. By [3.129]

   [s1]

   

   [s2]

   

   [s3]

   
3. By [3.131], we seek probabilities q₁ and q₂ such that

   [s4]

   

   By [3.132], those probabilities must satisfy

   [s5]

   

   Combining [s4] and [s5], we seek probabilities q₁ and q₂ such that

   [s6]

   

   This is a two-dimensional non-linear equation. It can be solved using Newton’s method. If you are using Microsoft Excel, you can simply use the solver function (see spreadsheet). The solution is

   [s7]

   

   Applying [3.133], we obtain the .25-quantile of X as –1.598.