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.