Exercise Solution 3.40

  1. By [3.128], μ = ξμ = 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.

 

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