SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

9
(a)
Pr{X 2 =1| X 1 =0} = Pr{X 2 =1,X 1 =0}
Pr{X 1 =0}
= 0.05
0.80 =0.0625
Pr{X 2 =1| X 1 =1} = 0.10
0.20 =0.50
Clearly X 1 and X 2 are dependent since Pr{X 2 =1| X 1 =0} ≠Pr{X 2 =1|
X 1 =1}.
{
100, a =0
(b) Let g(a) =
−20, a =1
E[g(X 2 ) | X 1 =1] = g(0) Pr{X 2 =0| X 1 =1}
+ g(1) Pr{X 2 =1| X 1 =1}
= 100(1 − 0.50) − 20(0.50)
= 40
10. ̂µ =2.75 ̂σ =0.064 ŝe = √ ̂σ 6
≈ 0.026
̂µ + ŝe = 2.776
̂µ − ŝe = 2.724
Assuming a mean of 2.776
Pr{X >2.90} = 1− Pr{X ≤ 2.90}
=
{ X − 2.776
1− Pr
≤
0.064
= 1− Pr{Z ≤ 1.938}
≈ 1 − 0.974 = 0.026
}
2.90 − 2.776
0.064
where Z is a standard-normal random variable.
Assuming a mean of 2.724
{
Pr{X >2.90} = 1− Pr Z ≤
}
2.90 − 2.724
0.064
= 1− Pr{Z ≤ 2.75} ≈1 − 0.997 = 0.003