# Exercise Solution 3.3

- By [3.4]
[s1]

[s2]

[s3]

[s4]

[s5]

- By [s5] and [3.7],
[s6]

[s7]

[s8]

[s9]

[s10]

[s11]

[s12]

- The standard deviation of a random variable is simply the square root of its variance:
[s13]

- 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]

- 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]

- 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. - A .875-quantile is any value
*z*for which:[s32]

This equation has the single solution z = 2.75.