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.