Answers to Selected Exercises

1.10.

$11121314212223243132333441424344$
$121314212324313234414243$
$11121314222324333444$
$121314232434$

1.11. 0.4162

1.12.

0.2070
0.1859

1.14.

0.9956
No

1.15. 0.2635

1.16. 0.1633.

1.23.

$1 4$
$1 221$
$4 17$
$1 52$

1.25. 0.000547

1.26.

$1 1296$
$5 54$
$5 3888$

1.28.

0.2937
0.7063

1.30.

$189 625$
$1 10000$

1.31.

$1 25$
$12 25$

2.28. Let $Y$ denote the number of defective chips in the sample

$Y k 10 k 90 5 k 1005$ for $k 012345$
$Y 0.5$ , $Y 0.432$ ,
$Y 0 0.416$ ,

2.29. Let $Y$ denote the number of women, so that $Z 10 Y$ , is the number of men.

$Y k 30 k 20 10 k 5010$ for $k 01 10$
$Y 6$ , $Y 1.959$ ,
$Z 4$ , $Z 1.959$ ,
$Y 0 Y 10 0.00294$ ,

2.30. $Y$ denote the number of tagged fish in the sample

$Y k 100 k 900 20 k 100020$ for $k 01 20$
$Y 2$ , $Y 196 111$
$Y 2 0.6108$ ,
$Y 2 0.6083$

2.31.

$Y k 10 k 0.4 k 0.6 10 k$ for $k 01 10$ .
$Y 4$ , $Y 2.4$
$Y 5 0.3669$

2.32. 20

2.33. 2000

2.34. 2000

2.35. Let $U$ denote the number of spades and $V$ the number of aces.

$U k 13 k 39 5 k 525$ for $k 01 5$ , $U 5 4$ , $U 235 272$
$V k 4 k 48 5 k 525$ for $k 01 4$ , $V 5 13$ , $V 940 2873$

2.36. Let $U$ denote the number of hearts and $V$ the number of honor cards.

$U k 13 k 39 13 k 5213$ for $k 01 13$ , $U 13 4$ , $U 507 272$
$V k 16 k 36 13 k 5213$ for $k 01 13$ , $V 4$ , $V 36 17$

3.15. Let $X$ , $Y$ , and $Z$ denote the number of republicans, democrats, and independents, respectively, in the sample.

$X i Y j Z k 40 i 35 j 25 k 10010$ where $i$ , $j$ , and $k$ are nonnegative integers with $i j k 10$ .
$X 4$ , $Y 3.5$ , $Z 2.5$
$X 2.1818$ , $Y 2.0682$ , $Z 1.7045$
$X Y 1.6364$ , $X Z 0.9091$ , $Y Z 0.7955$
0.2474

3.16. Let $X$ , $Y$ , $Z$ , $U$ , and $V$ denote the number of spades, hearts, diamonds, red cards, and black cards, respectively, in the hand.

$X i Y j Z k 13 i 13 j 13 k 13 13 i j k 5213$ where $i$ , $j$ , and $k$ are nonnegative integers with $i j k 13$ .
$X i Y j 13 i 13 j 26 13 i j 5213$ where $i$ and $j$ are nonnegative integers with $i j 13$ .
$X i 13 i 39 13 i 5213$ for $i 01 13$ .
$U i V j 26 i 26 j 5213$ where $i$ and $j$ are nonnegative integers with $i j 13$ .

3.17. Let $X$ , $Y$ , and $U$ denote the number of spades, hearts, and red cards, respectively, in the hand.

$X 13 4$ , $X 507 272$
$X Y -169 272$ ,
$U 13 2$ , $U 169 272$

3.18. Let $X$ , $Y$ , and $Z$ denote the number of spades, hearts, and diamonds respectively, in the hand.

$X i Y j Z 4 13 i 13 j 22 9 i j 489$ where $i$ and $j$ are nonnegative integers with $i j 9$ .
$X i Y 3 Z 2 13 i 34 8 i 478$ for $i 01 8$ .

4.16.

$X 53 k k 1 2 25 k 2 255$ for $k 34 22$
$X 53 13$
$X 53 125 7$ ,

4.17. 1437

4.19. 2322

5.29.

$k$ 0 1 2 3 4 5

$b 5 k$ 44 45 20 10 0 1
$k$ 0 1 2 3 4 5

$N 5 k$ 0.3667 0.3750 0.1667 0.0833 0 0.0083
Covariance: $1 100$ , correlation $1 16$

$k$	0	1	2	3	4	5
$b 5 k$	44	45	20	10	0	1

$k$	0	1	2	3	4	5
$N 5 k$	0.3667	0.3750	0.1667	0.0833	0	0.0083

5.30.

$k$	$N 10 k$
0	0.3678794
1	0.3678791
2	0.1839409
3	0.0613095
4	0.0153356
5	0.0030555
6	0.0005208
7	0.0000661
8	0.0000124
9	0
10	0.0000003

$N 10 1$ , $N 10 1$
$N 10 3 0.0803$

5.31.

See 5.30 (a)

$k$	$N k$
0	0.3678794
1	0.3678794
2	0.1839397
3	0.0613132
4	0.0153283
5	0.0030657
6	0.0005109
7	0.0000730
8	0.0000091
9	0.0000014
10	0.0000001

6.7. $5 18$

6.9. $89 144$

6.11. $189 625$

6.13. $n 9$

6.27.

$V 36530 j 30 j k 0 j 1 k j k j k 365 30$ for $j 12 30$
$V 36530 28.8381$
$V 36530 1.0458$
$V 36530 28 0.89767$

6.29.

$V 610 j 10 j k 0 j 1 k j k j k 6 10$ for $j 12 6$
$V 610 5.0310$
$V 610 0.5503$
$V 610 4 0.22182$

6.31.

$U 1015 j 15 j k 0 10 j 1 k 10 j k 1 j k 10 15$ for $j 01 9$
$V 1015 2.0589$
$V 1015 0.9864$
$V 1015 3 0.3174$

6.33.

$j$	1	2	3
$V 43 j$	$1 16$	$9 16$	$6 16$

$V 43 1 1 16$
$V 43 37 16$
$V 43 0.6830$

6.34.

$j$	1	2	3	4	5
$V 105 j$	$1 10000$	$927 2000$	$9 50$	$63 127$	$189 625$

$V 105 4.095$
$V 105 0.727$

7.12. Let $W W 36540$ denote the sample size.

$W n 36439 j 039 1 j 39 j 39 j 365 n 1$ for $n 4041$
$W 42.3049$
$W 2.4878$
$t W i 140 366 i 365 i 1 t$ for $t 365 39$

7.13. Let $W W 66$ denote the number of throws.

$W n j 05 1 j 5 j 5 j 6 n 1$ for $n 67$
$W 14.7$
$W 38.99$
$W 10 0.7436$

7.14. Let $W W 1010$ denote the number of boxes purchased.

$W n j 09 1 j 9 j 9 j 10 n 1$ for $n 1011$
$W 29.2897$
$W 125.6871$
$W 15 0.04595$

Answers to Selected Exercises

12. Finite Sampling Models

1. Introduction

2. The Hypergeometric Distribution

3. The Multivariate Hypergeometric Distribution

4. Order Statistics

5. The Matching Problem

6. The Birthday Problem

7. The Coupon Collector Problem