Exercise Solution 3.3

  1. By [3.4]

    [s1]

    [s2]

    [s3]

    [s4]

    [s5]

  2. By [s5] and [3.7],

    [s6]

    [s7]

    [s8]

    [s9]

    [s10]

    [s11]

    [s12]

  3. The standard deviation of a random variable is simply the square root of its variance:

    [s13]

  4. By [3.8]

    [s14]

    We obtain μ = 2 and σ = 1/ from items (a) and (c) above.

    [s15]

    [s16]

    [s17]

    [s18]

    [s19]

    [s20]

    Accordingly

    [s21]

  5. By [3.9]

    [s22]

    We obtain μ = 2 and σ = 1/ from items (a) and (c) above.

    [s23]

    [s24]

    [s25]

    [s26]

    [s27]

    [s28]

    Accordingly

    [s29]

  6. To determine the .10-quantile of Z, we construct the CDF Φ of Z. Formally, we do so by integrating its PF ϕ, which is given by [3.13]. However, since ϕ is so simple, Φ is easily obtained by inspection:

    [s30]

    A .10-quantile is any value z for which:

    [s31]

    This equation has the single solution z = 1.2.

  7. A .875-quantile is any value z for which:

    [s32]

    This equation has the single solution z = 2.75.

 

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