# Exercise Solution 3.2

- By [3.3],
[s1]

[s2]

[s3]

- By [s3] and [3.7],
[s4]

[s5]

[s6]

[s7]

- The standard deviation of a random variable is simply the square root of its variance:
[s8]

- 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]

- 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]

- 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. - 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*.