Answers to Selected Exercises

]> Answers to Selected Exercises

2. The Exponential Distribution

2.16.

$G t 1 s t$ , $g t s s t$ for $t 0$ , where $s i 1 n r i$
$H t i 1 n 1 r i t$ for $t 0$
$h t n 1 r t n 1 r r t$ for $t 0$

2.21. Let $X$ denote the call length.

$2 X 4 0.4237$
$Q 1 1.4384$ , $Q 2 3.4657$ , $Q 3 6.9315$ , $Q 3 Q 1 5.4931$

2.22. Let $T$ denote the lifetime

$T 2000 0.1353$
$Q 1 287.682$ , $Q 2 693.147$ , $Q 3 1386.294$ , $Q 3 Q 1 1098.612$

2.23. Let $T$ denote the time between requests.

$T 0.5$ , $T 0.5$
$T 0.5 0.6321$
$Q 1 0.1438$ , $Q 2 0.3466$ , $Q 3 0.6931$ , $Q 3 Q 1 0.5493$

2.24. Let $X$ denote the lifetime.

$r 0.02231$
$X 44.814$ , $X 44.814$
$Q 1 12.8922$ , $Q 2 31.0628$ , $Q 3 62.1257$ , $Q 3 Q 1 49.2334$

2.25. Let $X$ denote the position of the first defect.

$r 0.01$
$X 200 X 150 0.3935$
$X 100$
$Q 1 28.7682$ , $Q 2 69.3147$ , $Q 3 138.6294$ , $Q 3 Q 1 109.8612$

3. The Gamma Distribution

3.4. 0.1991

3.5. 0.1746

3.10. 2, 0.6325

3.11. $r 1 10$ , $k 4$

3.15.

0.5302
0.4871

3.16. 0.5752

3.20. $r 6.67$ hits per minute

4. The Poisson Distribution

4.6. 0.7798

4.7. 0.8153

4.12. 32, 5.657

4.19.

0.6157
0.6025

4.20. 0.8818

4.23. 0.6

4.26. 0.9452

4.30. $r 5.7$ per minute

5. Splitting a Poisson Process

5.6. 0.5814

5.11.

6. Bernoulli Trials and the Poisson Process

6.9.

0.8245
0.8153

6.10. 0.7350

6.11.

0.2063
0.2149
0.2146

6.12.

0.6066
0.5837
0.6247

6.13.

0.1227
0.0803

7. Higher Dimensional Poisson Processes

7.4.

0.4562
2.5, 1.581

7.5.

0.2426
24, 4.899

7.6.

$r 80$ per square mile
0.0171

7.13. 0.0491

7.16. 0.2146

7.18.

Mild: 7.854, 2.802; Moderate: 4.712, 2.171; Severe: 3.142, 1.772
0.7762