Exercise Solution 3.2

  1. By [3.3],

    [s1]

    [s2]

    [s3]

  2. By [s3] and [3.7],

    [s4]

    [s5]

    [s6]

    [s7]

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

    [s8]

  4. By [3.8],

    [3.8]

    we obtain σ = from item (c) above. Applying [3.5] with function f(y) = (y – μ)3, we obtain:

    [s9]

    [s10]

    [s11]

    Accordingly,

    [s12]

  5. By [3.9],

    [3.9]

    we obtain σ = from item (c) above. Applying [3.5] with function f(y) = (y – μ)4, we obtain:

    [s13]

    [s14]

    [s15]

  6. To determine the .10-quantile of Y, we construct the CDF Φ of Y from its PF ϕ, which is given by [3.12]. We obtain:

    [s16]

    A .10-quantile is any value y for which Φ(y) = .10. In [s16], we see that there is no such value. Accordingly, a .10-quantile of Y does not exist.

  7. A .875 quantile is any value y for which Φ(y) = .875. In [s16], we see that all values y in the interval [2, 3) satisfy this condition, so they are all .875 quantiles of Y.

 

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