##### Document Text Contents

Page 2

1.1.1.

a) The sample points are

� � � � � � � � � � � �

and the elementary events are� � � � � � � � � � � � � � � � � � � �

b) The event that corresponds to the statement � at leastone tail isobtained� is � � � � � � � � � � .

c) The event that corresponds to � at mostone tail isobtained� is � � � � � � � � � � �

1.1.3.

a) Four different sample spaces to describe three tossesof a coin are:�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� � � �

� � � � � � � � �� �

�

an even # of

� �

s, an odd #of

� �

s

� �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

where the fourth let-

ter is to be ignored in each sample point.

b) For

�

the event corresponding to the statement � at most one tail is obtained in three

tosses� is � � � � � � � � � � � � � � � � � . For � � it is � � � � � � � � and in � � it is not possible

to � nd such an event. For � � the event corresponding to the statement � at most one tail is

obtained in the � rst three tosses� is� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

c) It isnot possible to � nd anevent corresponding to the statement � at mostone tail isobtained

in three tosses� inevery conceivable sample space for the tossingof three coins,because some

sample spaces are toocoarse, that is, the sample points that contain this outcome also contain

oppositeoutcomes. For instance, in

� �

above, the sample point � an even # of � �s� contains

the outcomes

� � � � � � � � � � �

for which our statement is true and the outcome

� � � �

for which it is not true.

1.1.5.

In the 52-element sample space for the drawing of a card

a) the events corresponding to the statements �

� An Ace or a red King is drawn,� and �

� The card drawn isneither red,nor odd,nor a face card� are �

� � � � � � � � � � � � � � � � � � �

and �

� � � � � � � � � � � � ! " � � # $ � � $ � � $ � $ � ! " $ % , and

b) statements corresponding to the events& ' ( ) * � + * � , * � - * %

, and . ' ( # � � � � � � � � � � ! " � � # $ � � $ � � $ � $ � ! " $ % are / '0

The Aceof hearts or aheart face card is drawn,

1

and 2 ' 0 An even numberedblack card is

drawn.

1

1.1.7.

Three possible sample spaces are:$ 3 ' (

The 365 daysof theyear

% �

1

Page 29

4.4.9.

Clearly, using the numbered regions from the Figure, � � � & � % & � ' � � � " � % & � � P� � �� & � � � % & � ' P� � � � " � � � % & � � P� �

� � � �

' P�

� �

� P� � �

� P� � " � � �� & � � � % & � and � � � & � % " � ' � � � " � % " � � P� � � � & � � � % " � ' P� � � � " � � � % " � �P� � � �

' P� �

� P� �

� P� � " � � � � & � � � % " � , Thus, � � � & � % & � ' � � � " � % & � �� � � � � � � � � � � � � � � � � � P� � � � � � P� � � � P� � � � � � � � ! � � � � P� � � � � � � � ! � � � �

P� � � � P� � � � � � � � � � � ! � � � "

4.5.1.�

and

!

are not independent: For instance, # � $ � � � � P� � � $ � ! � � � � � % &' � � % &% � � ( )% �� * (( � �� +, � � # - . / 0 1 . 2 34 0 . 3 56 0. 7 66 0 1 8 9: ; < and = > . ? 0 1 . 2 32 0 . 3 52 0. 7 66 0 1 8 :: ; @ Now, 8 9: ; A B CC D EF B CG B H

4.5.3.I

and J arenot independent: By Example 4.4.5,I and J areboth uniform on the intervalK L M N O

and, by Example 4.5.2,P I M J Q is then uniform on the unit square and not on R H

4.5.5.

1. By the deS nition of indicators, T U V W X Y Z [ \ X ] ^ _ ` By the dea nition of

intersection, X ] ^ _ \ W X ] ^ and X ] _ Y b and, by the dea nition of indicators, W X ] ^

and X ] _ Y \ W c U W X Y Z [ and c V W X Y Z [ Y ` Since [ d [ Z [ and [ d e Z e d e Z e b

clearly, W c U W X Y Z [ and c V W X Y Z [ Y \ c U W X Y c V W X Y Z [ ` Now, by the transitivity

of equivalencerelations, c U V W X Y Z [ \ c U W X Y c V W X Y Z [ b which is equivalent toc U V Z c U c V `

2. By dea nition, c U f g h i j k l m i n o p q r Similarly, s t h i j u s g h i j v s t g h i j kl m i n o p q w becauseo p q k o q p o q p o q w where the three constituents

are disjoint, and s t h i j u s g h i j v s t g h i j is l u x v x k l for i n o q w

is x u l v x k l for i n o q , and l u l v l k l for i n o q r Thus, by

transitivity, s t f g h i j k l m s t h i j u s g h i j v s t g h i j k l w which is equivalent to

28

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s t f g k s t u s g v s t g r

3. By de

�

nition, � � and � � are independent if and only if P � � � � � � � � � � �

�

P� � � � � �

P� � � � �

for � � � � � � � �

By the de� nition of indicators, this relation

holds if and only if P � � �

� P� �

P� �

�P� � � � � P� � � P� � � �P� � � � � P� � � P� � � �

and P� � � � � P� � � P� � � all hold. By the de� nition of independenceof two events and

the resultof Exercise 3.3.5, the last four equations are equivalent to the independenceof� and � � Thus,by transitivity, � � and � � are independent if and only if � and � are.

4.5.7.

If � and � denote the arrival timesof Alice and Bob, respectively, then they will meet if

and only if � ! " � # $ or, equivalently, ! $ # " # % $ & Now, ' ( " ) is uniformly

distributedon the square * + ( , - . * + ( , - ( and the above condition is satis/ edby the points of

the shadedregion, whose area is 0 1 ! the areaof the two triangles 2 $ , ! 3 2 4 & Thus,

P' 5 and 6 meet) 2 78 9 &

29

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Thus,EA B G � M � K I J R � BK I J R � � N � M I J J � BM I J J � � K � K I M Q � BK I M Q � � K � M I S Q � BM I S Q � � O � M I N K � BM I N K� � M � K I M Q � BK I M Q � � J � J I H O � BJ I H O � � P P � P P I H P � BP P I H P � � J � Q I K S � BQ I K S� � P P � J I O S � BJ I O S � � J � P H I R N � BP H I R N � � Q � Q I K S � BQ I K S G P I P N

Thenumber of degreesof f reedom is � N � P � � S � P � � M and so, from a table, [email protected] A B� D EA B F GH I J M � which givesoverwhelming support to thehypothesisof independence.

7.6.1.

We take� �

� �

� � � � and � � �

� � � From the data,�� � � � � � � � � ! ! " # $ % & '# ! ! ( ) * + , -. / 0 1 2 3 4 0 5 2 10 2 6 1 7 8 9 : ; and so P< = > 8 9 : ; ? 7 @ 9 @ @ 8 A 9

7.6.3.

By the de

B

nition of C D E F and the independenceof the chi-squarevariables involved,G < C D E F ? H G I J K LMN K LO P Q JN R S T UV W R X YZ [\ ] ^ Now, _ ` a bc d e f and _ g hi jk ] el mn op q r st u v w x v y z {| }~ �� � � � � � � � � � � �� � � � � �� � � � � � � � � � � � � � � � � � � ¡ �� ¢ £ ¤ ¥ ¦ � � � � § ¨© ª « if ¬ ® ¯

Hence,° ± ² ³ ´ © µ § ©© ª « ¯

7.7.1.

Use the large sample formula ¶ ± · © ¸ ¹ º » ¼ ½ ¾ ¿À Á Â Ã Ä Å Æ À Ç Â È É Ê Ë Ì Í Ì Î From Ï ÐÑ Ò Ó Ô Ñ

we getÏ ÐÕ Ö Ö Ò Ó × Ø × Ù Ú and so Ò Û Ü Ý Þ ß à Ý Thus,á â ã ä å æ ä ç Û è é êë ì í î ï ð ñ ë ò í ó ô õ ö ÷ ø ù ú û ü ÷ ýþ ÿ � � �

and, this valuebeing fairly large,we accept� � ÿ

7.7.3.

Since � � � �

� �

�

� hasonly a � nite number of values, it does assume its supremum at

somevaluesof � � that is, its supremum is its maximum. Also, since � � � � � and � � � � � are

right-continuous step functions with jumps at the � � , � � � � � � � � � � � � � � � � is assumed at

every point of an interval � � � � � � and, in particular, at � � !

7.7.5.

Use the large sample formula P" # $ % & ' ( $ ) %$ % * + , - ./ 0 1 2 3 4 5 6 7 8 9 : ; < = > = ? From @ A B C DB D EF B D we getG A H I I C J I IH I I K L M M N O P O Q R and so S T O P Q U V P Thus, PW X Y Z [ \ Y Z ] ^_ ` ab c d e f g h b i d j k l m n o p q r s n t u v w x v We accepty z v

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1.1.1.

a) The sample points are

� � � � � � � � � � � �

and the elementary events are� � � � � � � � � � � � � � � � � � � �

b) The event that corresponds to the statement � at leastone tail isobtained� is � � � � � � � � � � .

c) The event that corresponds to � at mostone tail isobtained� is � � � � � � � � � � �

1.1.3.

a) Four different sample spaces to describe three tossesof a coin are:�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� � � �

� � � � � � � � �� �

�

an even # of

� �

s, an odd #of

� �

s

� �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

where the fourth let-

ter is to be ignored in each sample point.

b) For

�

the event corresponding to the statement � at most one tail is obtained in three

tosses� is � � � � � � � � � � � � � � � � � . For � � it is � � � � � � � � and in � � it is not possible

to � nd such an event. For � � the event corresponding to the statement � at most one tail is

obtained in the � rst three tosses� is� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

c) It isnot possible to � nd anevent corresponding to the statement � at mostone tail isobtained

in three tosses� inevery conceivable sample space for the tossingof three coins,because some

sample spaces are toocoarse, that is, the sample points that contain this outcome also contain

oppositeoutcomes. For instance, in

� �

above, the sample point � an even # of � �s� contains

the outcomes

� � � � � � � � � � �

for which our statement is true and the outcome

� � � �

for which it is not true.

1.1.5.

In the 52-element sample space for the drawing of a card

a) the events corresponding to the statements �

� An Ace or a red King is drawn,� and �

� The card drawn isneither red,nor odd,nor a face card� are �

� � � � � � � � � � � � � � � � � � �

and �

� � � � � � � � � � � � ! " � � # $ � � $ � � $ � $ � ! " $ % , and

b) statements corresponding to the events& ' ( ) * � + * � , * � - * %

, and . ' ( # � � � � � � � � � � ! " � � # $ � � $ � � $ � $ � ! " $ % are / '0

The Aceof hearts or aheart face card is drawn,

1

and 2 ' 0 An even numberedblack card is

drawn.

1

1.1.7.

Three possible sample spaces are:$ 3 ' (

The 365 daysof theyear

% �

1

Page 29

4.4.9.

Clearly, using the numbered regions from the Figure, � � � & � % & � ' � � � " � % & � � P� � �� & � � � % & � ' P� � � � " � � � % & � � P� �

� � � �

' P�

� �

� P� � �

� P� � " � � �� & � � � % & � and � � � & � % " � ' � � � " � % " � � P� � � � & � � � % " � ' P� � � � " � � � % " � �P� � � �

' P� �

� P� �

� P� � " � � � � & � � � % " � , Thus, � � � & � % & � ' � � � " � % & � �� � � � � � � � � � � � � � � � � � P� � � � � � P� � � � P� � � � � � � � ! � � � � P� � � � � � � � ! � � � �

P� � � � P� � � � � � � � � � � ! � � � "

4.5.1.�

and

!

are not independent: For instance, # � $ � � � � P� � � $ � ! � � � � � % &' � � % &% � � ( )% �� * (( � �� +, � � # - . / 0 1 . 2 34 0 . 3 56 0. 7 66 0 1 8 9: ; < and = > . ? 0 1 . 2 32 0 . 3 52 0. 7 66 0 1 8 :: ; @ Now, 8 9: ; A B CC D EF B CG B H

4.5.3.I

and J arenot independent: By Example 4.4.5,I and J areboth uniform on the intervalK L M N O

and, by Example 4.5.2,P I M J Q is then uniform on the unit square and not on R H

4.5.5.

1. By the deS nition of indicators, T U V W X Y Z [ \ X ] ^ _ ` By the dea nition of

intersection, X ] ^ _ \ W X ] ^ and X ] _ Y b and, by the dea nition of indicators, W X ] ^

and X ] _ Y \ W c U W X Y Z [ and c V W X Y Z [ Y ` Since [ d [ Z [ and [ d e Z e d e Z e b

clearly, W c U W X Y Z [ and c V W X Y Z [ Y \ c U W X Y c V W X Y Z [ ` Now, by the transitivity

of equivalencerelations, c U V W X Y Z [ \ c U W X Y c V W X Y Z [ b which is equivalent toc U V Z c U c V `

2. By dea nition, c U f g h i j k l m i n o p q r Similarly, s t h i j u s g h i j v s t g h i j kl m i n o p q w becauseo p q k o q p o q p o q w where the three constituents

are disjoint, and s t h i j u s g h i j v s t g h i j is l u x v x k l for i n o q w

is x u l v x k l for i n o q , and l u l v l k l for i n o q r Thus, by

transitivity, s t f g h i j k l m s t h i j u s g h i j v s t g h i j k l w which is equivalent to

28

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s t f g k s t u s g v s t g r

3. By de

�

nition, � � and � � are independent if and only if P � � � � � � � � � � �

�

P� � � � � �

P� � � � �

for � � � � � � � �

By the de� nition of indicators, this relation

holds if and only if P � � �

� P� �

P� �

�P� � � � � P� � � P� � � �P� � � � � P� � � P� � � �

and P� � � � � P� � � P� � � all hold. By the de� nition of independenceof two events and

the resultof Exercise 3.3.5, the last four equations are equivalent to the independenceof� and � � Thus,by transitivity, � � and � � are independent if and only if � and � are.

4.5.7.

If � and � denote the arrival timesof Alice and Bob, respectively, then they will meet if

and only if � ! " � # $ or, equivalently, ! $ # " # % $ & Now, ' ( " ) is uniformly

distributedon the square * + ( , - . * + ( , - ( and the above condition is satis/ edby the points of

the shadedregion, whose area is 0 1 ! the areaof the two triangles 2 $ , ! 3 2 4 & Thus,

P' 5 and 6 meet) 2 78 9 &

29

Page 57

Thus,EA B G � M � K I J R � BK I J R � � N � M I J J � BM I J J � � K � K I M Q � BK I M Q � � K � M I S Q � BM I S Q � � O � M I N K � BM I N K� � M � K I M Q � BK I M Q � � J � J I H O � BJ I H O � � P P � P P I H P � BP P I H P � � J � Q I K S � BQ I K S� � P P � J I O S � BJ I O S � � J � P H I R N � BP H I R N � � Q � Q I K S � BQ I K S G P I P N

Thenumber of degreesof f reedom is � N � P � � S � P � � M and so, from a table, [email protected] A B� D EA B F GH I J M � which givesoverwhelming support to thehypothesisof independence.

7.6.1.

We take� �

� �

� � � � and � � �

� � � From the data,�� � � � � � � � � ! ! " # $ % & '# ! ! ( ) * + , -. / 0 1 2 3 4 0 5 2 10 2 6 1 7 8 9 : ; and so P< = > 8 9 : ; ? 7 @ 9 @ @ 8 A 9

7.6.3.

By the de

B

nition of C D E F and the independenceof the chi-squarevariables involved,G < C D E F ? H G I J K LMN K LO P Q JN R S T UV W R X YZ [\ ] ^ Now, _ ` a bc d e f and _ g hi jk ] el mn op q r st u v w x v y z {| }~ �� � � � � � � � � � � �� � � � � �� � � � � � � � � � � � � � � � � � � ¡ �� ¢ £ ¤ ¥ ¦ � � � � § ¨© ª « if ¬ ® ¯

Hence,° ± ² ³ ´ © µ § ©© ª « ¯

7.7.1.

Use the large sample formula ¶ ± · © ¸ ¹ º » ¼ ½ ¾ ¿À Á Â Ã Ä Å Æ À Ç Â È É Ê Ë Ì Í Ì Î From Ï ÐÑ Ò Ó Ô Ñ

we getÏ ÐÕ Ö Ö Ò Ó × Ø × Ù Ú and so Ò Û Ü Ý Þ ß à Ý Thus,á â ã ä å æ ä ç Û è é êë ì í î ï ð ñ ë ò í ó ô õ ö ÷ ø ù ú û ü ÷ ýþ ÿ � � �

and, this valuebeing fairly large,we accept� � ÿ

7.7.3.

Since � � � �

� �

�

� hasonly a � nite number of values, it does assume its supremum at

somevaluesof � � that is, its supremum is its maximum. Also, since � � � � � and � � � � � are

right-continuous step functions with jumps at the � � , � � � � � � � � � � � � � � � � is assumed at

every point of an interval � � � � � � and, in particular, at � � !

7.7.5.

Use the large sample formula P" # $ % & ' ( $ ) %$ % * + , - ./ 0 1 2 3 4 5 6 7 8 9 : ; < = > = ? From @ A B C DB D EF B D we getG A H I I C J I IH I I K L M M N O P O Q R and so S T O P Q U V P Thus, PW X Y Z [ \ Y Z ] ^_ ` ab c d e f g h b i d j k l m n o p q r s n t u v w x v We accepty z v

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