SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
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60 CHAPTER 6. DISCRETE-TIME PROCESSES<br />
⎛<br />
P =<br />
⎜<br />
⎝<br />
1 0 0 0 ··· 0<br />
b(1, 1) b(1, 0) 0 0 ··· 0<br />
b(2, 2) b(2, 1) b(2, 0) 0 ··· 0<br />
b(3, 3) b(3, 2) b(3, 1) b(3, 0) ··· 0<br />
··· ··· ··· ···<br />
. . . ···<br />
b(k, k) b(k, k − 1) b(k, k − 2) b(k, k − 3) ··· b(k, 0)<br />
⎞<br />
⎟<br />
⎠<br />
(b) p (m−1)<br />
k0<br />
With k =6,p=0.1 <strong>and</strong>m =12weobtainp (11)<br />
60 ≈ 0.104.<br />
(c) ∑ k<br />
l=0 lp (m)<br />
kl<br />
With k =6,p=0.1 <strong>and</strong>m =12weobtain ∑ 6<br />
l=0 lp (12)<br />
6l ≈ 1.7.<br />
(d) For each prisoner, the probability that they have not been paroled by m months<br />
is<br />
q =(1− p) m<br />
There<strong>for</strong>e, 1 − q is the probability they have been paroled be<strong>for</strong>e m months.<br />
The prison will close if all have been paroled. Since the prisoners are independent<br />
Pr{all paroled be<strong>for</strong>e m} =(1− q) k =(1− (1 − p) m ) k<br />
20. (a) M = {1, 2, 3} ≡{good <strong>and</strong> declared good, defective but declared good, defective<br />
<strong>and</strong> declared defective}<br />
n ≡ number of items produced<br />
21. (a)<br />
(b) π 2 =0.0006<br />
P =<br />
⎛<br />
⎜<br />
⎝<br />
0.995 0.005(0.06) 0.005(0.94)<br />
0.495 0.505(0.06) 0.505(0.94)<br />
0.495 0.505(0.06) 0.505(0.94)<br />
M = {1, 2, 3, 4}<br />
= {(no, no), (no, accident), (accident, no), (accident, accident)}<br />
n counts number of years<br />
⎞<br />
⎟<br />
⎠