UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO FACULT FACULTAD DE ESTUDIOS E STUDIOS SUPERIORES SUPERIORE S ACATLÁN ACATLÁN LICENCIATURA EN ECONOMÍA
SEGUNDA PRÁCTICA DE ESTADÍSTICA
ESTADÍSTICA I GALÁN FIGUEROA JAVIER
GARCÍA PRIETO ERIC GUILLERMO KAHZAIN GUTIERREZ MORENO CESAR LÓPEZ BECERRA EDGAR DANIEL
2.1 S!"#$% &' #% (%)*+*% ,"#-*'#' "/ 0*"/ ' '%'/ *(''#-'/ 3 '/-%)"/ *#-''/%"/ '# '+ $4#'" ' '/-"/ #*5"/. D'#"-')"/ ,"# F &' #% 0*% '/ )' 3 M &' '+ 0*" '/ 0")6' 3 '#"-' ,"# # !%7 !" '')!+"7 F M 7 &' '+ 0*" ' )%3" '% '/ +% #*5% 3 '+ )8/ "9'# '/ '+ #*5". H%3 ,%-" !#-"/ '# '+ ,"##-" S ' !"/*6+'/ "6/'9%,*"#'/: S ={ FF , FM , MF , MM } D'#"-' ,"# A '+ /6,"##-" ' !"/*6*+*%'/ &' #" ,"#-'#$% 0")6'/; B7 '+ /6,"##-" &' ,"#-*'#' "/ 0")6'/; 3 C7 '+ /6,"##-" &' ,"#-'#$% %+ )'#"/ # 0")6'. I#*&' +"/ '+')'#-"/ ' A7 B7 C7 A
A ?@ A ∪B= >FF7 MM@ A∩C= >FF@ A∪C = >MM7 MF7 FM@ B∩C = >MM@ B∪C = >MM7 MF7 FM@ C∩B= >MM@ ´ = { FF , FM , MF } C∩B 2.2 S!"#$% &' A 3 B /"# "/ '9'#-"/. E/,*6% +%/ '!'/*"#'/ &' ,"#-'#$%# #*"#'/7 *#-'/',,*"#'/ 3 ,")!+')'#-"/ &' '/,*6%# +" /*$*'#-': % O,'# %)6"/ '9'#-"/. 6 A+ )'#"/ #" ",'. , N*#$#" ",'. E%,-%)'#-' #" ",'. % AB 6 A
´ B
= A<B∪
´ B
6 S* B ⊂ A '#-"#,'/ A = B ∪A<
= A
.
b.
´ B∪ A< B
= B< A∪B<
´ B
= B< A = A .
, A')8/7 ')'/-' &' A A7 B7 AB7 O@ 6 H%$% /" ' +% *#(")%,*# %% %#-'/ !%% %/*$#% !"6%6*+*%'/ % ,%% #" ' +"/ '9'#-"/ /*)!+'/. b. P >A@ = ?.17 P >B@ = ?.1?7 P >AB@ = ?.?7 P >O@ = ?.. , C8+ '/ +% !"6%6*+*% ' &' +% !'/"#% /'+',,*"#%% %+ %% -'#$% -*!" ' /%#$' A " -*!" AB c. P >A7 B@ = P >A@ P >B@ = ?.1 2.12 U# 9'0,+" &' ++'$% % # ,,'" !'' % 9'+-% % +% '',0%7 % +% *&*'% " ,"#-*#% ' ('#-'. E+ '!'*)'#-" ,"#/*/-' '# "6/'9% '+ )"9*)*'#-" ' # /"+" 9'0,+" !" '+ ,,'". a I#*&' '+ '/!%,*" )'/-%+ !%% '/-' '!'*)'#-". b S!"#*'#" &' -""/ +"/ !#-"/ )'/-%+'/ /"# *$%+)'#-' !"6%6+'/7
'#,'#-' +% !"6%6*+*% ' &' '+ 9'0,+" 4 9'+-%. 2.1 L"/ '/-%"#*'#/'/ !''# /' 6%/-%#-' //!*,%,'/7 '# '/!',*%+ ,%#" /' -%-% ' ,"#/!*%,*"#'/ '+ $"6*'#". S"6' +% !'$#-% ' /* +% F'% A4'% ' E/-%"/ U#*"/ -*'#' ",+-% +% !'6% ' +% '*/-'#,*% ' 9*% *#-'+*$'#-' '# "-"/ !+%#'-%/7 +%/ !"!",*"#'/ ' '/-%"#*'#/'/ ,"# "!*#*"#'/ &' 9%%# /' %# '# +% -%6+%. Opinión Muy probable Poco Probable Improbable Otra
Proporción
?.2 ?.2 ?.? ?.12
S!"#$% &' /' /'+',,*"#% # '/-%"#*'#/' 3 &' /' '$*/-% / "!*#*#.
% C8+'/ /"# +"/ '9'#-"/ /*)!+'/ !%% '/-' '!'*)'#-" a. S= { M3 !"6%6+'=MP7 P"," !"6%6+'=PP7 I)!"6%6+'=I7 O-%=O@ 6 T""/ +"/ '9'#-"/ /*)!+'/ &' /-' *" '# '+ *#,*/" %7 /"# *$%+)'#-' !"6%6+'/ S* #" '/ %/7 ,8+'/ /"# +%/ !"6%6*+*%'/ &' '6'# %/*$#%/' % ,%% #" b. P MP = .27 P PP = .27 P I = .?7 P O = .12. , C8+ '/ +% !"6%6*+*% ' &' +% !'/"#% /'+',,*"#%% '#,'#-' %+ )'#"/ # !"," !"6%6+' &' +% F'% A4'% '/-4 ",+-%#" *#(")%,*# %,',% ' 9*% *#-'+*$'#-' '# "-"/ !+%#'-%/ c. P PP = P MP P PP = .. 2.30 T'/ 9*#"/ *)!"-%"/ 9%# % /' ,+%/*(*,%"/ ' )'#"/ % )8/ !" # '!'-"
'# 9*#"/. E/-" '/7 # 9*#" /'8 *'#-*(*,%" ,")" '+ )'"7 "-" ,")" '+ /'$#" )'" 3 '+ 9*#" '/-%#-' ,")" '+ !'". a D'/,*6% # !#-" )'/-%+ !%% '/-' '!'*)'#-". b I#*&' '+ '/!%,*" )'/-%+. c S!"#$% &' '+ '!'-" #" /%6' '# '%+*% #%% ' 9*#"/ 3 &' %+ %%
%/*$#% '+ +$% % +"/ -'/ 9*#"/. U#" ' +"/ 9*#"/ '/ ' ),0% )'" ,%+*% &' +"/ "-"/. C8+ '/ +% !"6%6*+*% ' &' '+ '!'-" #" ,+%/*(*&' '+ )'" 9*#" ,")" !'" &' '+ /'$#" )'" R= a) D'#"-%)"/ %+ !*)'" 9*#" ,")" 917 %+ S'$#" ,")" 92 3 %+ -','" ,")" 9. C%% !#-" ' )'/-% '/ # ,"##-" ' &' *#*,% '+ "'#. b) S= 917 927 97 917 97 927 927 917 97 927 97 917 97 917 927 97 927 91 c) P%% ,%% 9*#" 0%3 ,%-" !"/*6+'/ '#'/ '# +"/ &' #" '/ '+ +-*)". P" +" -%#-"7 +% !"6%6*+*% '/ ' 2Q. 2.33 L% O(*,*#% '+ C'#/" '!"-% &' '+ *#$'/" (%)*+*% )'*%#" !%% -"%/ +%/
(%)*+*%/ '# E/-%"/ U#*"/7 %#-' '+ %5" 2??7 (' 71. E/-" '/7 +% )*-% ' -"%/ +%/ (%)*+*%/ '/-%"#*'#/'/ -9" *#$'/"/ &' '6%/%6%# '/-% ,%#-*% 3 +% )*-% -9" *#$'/"/ *$%+'/ " )'#"'/ % '/-% ,%#-*%. S!"#$% &' /' 0%,' #% '#,'/-% % ,%-" (%)*+*%/ 3 &' ,%% #% ' '++%/ '9'+% /* / *#$'/" '6%/ +"/ 71 '# 2??. a I#*&' +"/ !#-"/ '+ '/!%,*" )'/-%+. b I'#-*(*&' +"/ '9'#-"/ /*)!+'/ '# ,%% #" ' +"/ '9'#-"/ /*$*'#-'/:
%+ )'#"/ "/ -9*'"# *#$'/"/ ' )8/ ' 71. • B: '%,-%)'#-' "/ -9*'"# *#$'/"/ ' )8/ ' 71. • C : '%,-%)'#-' #% -9" *#$'/"/ )'#"'/ " *$%+'/ % 71. c H%$% /" ' +% *#-'!'-%,*# %% !%% +% )'*%#% !%% %/*$#% !"6%6*+*%'/ % +"/ '9'#-"/ /*)!+'/ 3 '#,'#-' P A7 P B 3 P C . •
A:
R=a) D'(*#*'#" +"/ 2 */-*#-"/ '9'#-"/ ,")" A= *#$'/" (%)*+*% )%3" % 7 1 ' B= *#$'/" (%)*+*% *$%+ " )'#" % 71. L"/ !#-"/ /"#:
E1: AAAA E2: AAAB E: AABA E: ABAA E: BAAA E: AABB E: ABAB E: BAAB E: ABBA E1?: BABA E11: BBAA E12: ABBB E1: BABB E1: BBAB E1: BBBA E1: BBBB b) A = >E17 E27 7 E11@ B = >E7 E7 7 E11@ C = >E27 E7 E7 E@ c) S* P E = P N = .7 ,%% '+')'#-" '+ '/!%,*" )'/-% -*'#' #% !"6%6*+*% ' 1Q1. A/7 P A = 11Q17 P B = Q1 3 P C = Q1. 2.35 U#% +#'% %4'% -*'#' /'*/ 9'+"/ ' N'9% W" % C%+*("#*% 3 /*'-' 9'+"/
' C%+*("#*% % H%Y8* *%*"/. S* +"/ 9'+"/ /' 0%# ' 0%,' '# %/ /'!%%"/7 ,8#-"/ %'$+"/ *(''#-'/ ' 9'+"/ !'' "(',' +% +#'% %4'% ' N'9% W" % H%Y8* R = E+ #)'" -"-%+ ' 9'+"/ '/: = 2. 2.40 U#% )%,% ' %-")9*+ 9*'#' '# ,*#," '/-*+"/ *(''#-'/7 ,"# ,%-" -*!"/ '
)"-"7 ,"# "/ -*!"/ ' -%#/)*/*"#'/ 3 '# ",0" ,"+"'/. C8#-"/ %-"/ -'#% &' -'#' '# '*/-'#,*% # */-*6*" /* *#,+3'% #" !" ,%% ,")6*#%,*# ' '/-*+"7 )"-" 3 -%#/)*/*# b C8#-"/ -'#% &' -'#' '# '*/-'#,*% # ,'#-" ' */-*6,*# /* -""/ +"/ ,"+"'/ ' %-"/ /' -9*'%# !%% ,%% ,")6*#%,*# '+ *#,*/" % a
R= a) P" '$+% ' ,")6*#%,*"#'/7 '+ */-*6*" '6' %6%/-','/' ,"# ? %-"/ 3% &': 2 = ? b P%% &' ,%% #" ' '/-%/ ,")6*#%,*"#'/ '/-9*'% '# +"/ ,"+"'/7 '+ */-*6*" '6' %6%/-','/' ,"# 2? %-"/7 3% &': ? = 2? 2.43 U#% (+"-% ' #'9' -%*/ /' 0% ' '/!%,0% % -'/ %'"!'-"/ '# (")% -%+
&' -'/ 9%3%# %+ %'"!'-" A7 ,*#," %+ %'"!'-" B 3 #" %+ %'"!'-" C. E# ,8#-%/ (")%/ */-*#-%/ /' !'' +"$% '/-" R= 1 =? (")%/ */-*#-%/. 1 2.44 C"#/+-' '+ E',*,*" 2.. S!"#$% &' +"/ -%*/ /"# %/*$#%"/ %
%'"!'-"/ %+ %%. R= a) E+ #)'" ' )%#'%/ '# &' '+ -%* &' #','/*-% '!%%,*# !'' /' '#9*%" %+ %'"!'-" C '/: = )%#'%/. P" +" -%#-"7 +% !"6%6*+*% '/: Q? = 1Q. b) = 7 !" +" &'7 +% !"6%6*+*% ' &' ,%% %'"!'-" ',*6% #" ' +"/ -%*/ &' 2 '&*''# '!%%,*# '/ ' Q?= ?.?2
2.? L% !"6%6*+*% '/')!'5 # !%!'+ '# +% )%#*!+%,*# ' +% +"-'% '+ '/-%" ' P'##/3+9%#*% '+ 2 ' %6*+ ' 1? L"/ Á#$'+'/ T*)'/7 ' /'!-*')6' ' 1?. P%% '-')*#% ,%% #" ' +"/ $*-"/ '+ #)'" $%#%" ' -'/ $*-"/7 ,%% #" ' +"/ #)'"/ ?7 17 27...7 /' ,"+",% '# #% !'+"-% ' !*#$‒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
( )=
50 !
50
H%3
3
?"
=
(
= 17?? )%#'%/ ' '+'$* +"/ $%#%"'/.
)
3 ! 50 −3 !
C%% #" ' '++"/ '/ *$%+)'#-' !"6%6+'.
()=
4!
4
a) H%3
3
"
=
(
= )%#'%/ '# &' +"/ "$%#*%"'/ $%#'#
)
3 ! 4 −3 !
-""/ +"/ !')*"/. L% !"6%6*+*% '/ Q1??. b) H%3
( )( ) 4
46
2
1
=
(
4!
(
)
2 ! 4 −2 !
)(
46 !
(
)
1 ! 46−1 !
)=
2 )%#'%/ '# &' +"/
"$%#*%"'/ !''# $%#% "/ !')*"/ 3 #% ' +% "-%/ !'/"#%/ $%#' '+ -',' !')*". P" +" -%#-"7 +% !"6%6*+*% '/ 2Q1??. c) H%3
( )( )
1?Q1??.
4
46
1
2
=
(
4!
( −1 ) !
1! 4
)(
46 !
( −2 ) !
2 ! 46
)=
1?. L% !"6%6*+*% '/
( ) 46
d)
3
46 !
= " =
(
)
3 ! 46−3 !
= 171?. L% !"6%6*+*% '/ 11?Q1??
2.1 S* "/ '9'#-"/7 A 3 B7 /"# -%+'/ &' P A=.7 P B=.7 3 P A
( (
) ( ) = ( ) ) = ( ) ( ( )) =
P A ∩ B
.1
1
P A
.5
5
c) P AA ∪B =
P A
.5
P A ∪ B
.7
5
=
7
P A∪B = PA PB [ PA
(
P ( A ∩B ) P ( A ∩B )
e) P A
(
) = ) ) ) =
.1 .1
= 1
P ( A ∩ B
.1
P ( A ∪ B
.7
2.2 P%% ,*'-% !"6+%,*# ' ')!+'%"/7 '+ !",'#-%' &' %!'6% " '!'6% # '%)'# ' ,")!'-'#,*% '# # -%6%"7 *#*,%" ' %,'" ,"# '+ $4#'"7 ('"# ,")" /' 9' '# +% -%6+% /*$*'#-'. E/-" '/7 ' -"%/ +%/ !'/"#%/ &' -")%"# '+ '%)'#7 2\ '/-%6%# '# +% ,%-'$"% ' 0")6'[%!"6%"7 1\ '/-9*'"# '# +% ,%-'$"% ' )'['!"6%% 3 %/ /,'/*9%)'#-'. U# ')!+'%" 9% % /' /'+',,*"#%" %+ %% ' '/-% !"6+%,*#. S'% A '+ '9'#-" ' &' '+ ')!+'%" "6-'#$% #% ,%+*(*,%,*# ' %!"6%" '# '+ '%)'# 3 /'% M '+ '9'#-" ' &' /' /'+',,*"#' # 0")6'.
% L"/ '9'#-"/ A 3 M /"# *#'!'#*'#-'/ 6 L"/ '9'#-"/ Ᾱ 3 F /"# *#'!'#*'#-'/
! 60
a) T'#$% '# ,'#-% &' P A =
100
= ?. 3 P A M = ?.2 Q ?. = ?.. P" +"
-%#-"7 A 3 M /"# *#'!'#*'#-'/. 40
b) S*)*+%)'#-'7 P Ᾱ =
100
= ?. 3 P Ᾱ F = ?.2 Q ?. = ?. =7 !" +" &' Ᾱ 3
F /"# *#'!'#*'#-'/. 2. G'$" M'#'+ (' # )"#' &'7 '# 17 /$** #% -'"% ' +% 0''#,*% 6%/%% '# +% ,*'#,*% ' +% $'#4-*,%. É+ *'#-*(*, *#*9*"/ 0'-'",*$"-"/ !%% ,"+" ' (+" &' -'#%# "/ %+'+"/ # = %+'+" ,"+" 6+%#," ','/*9" 3 # R= %+'+" ,"+" "" ")*#%#-'. C%#" '/-"/ *#*9*"/ /' %!%'%6%#7 /' "6/'9%6% &' Q ' +% '/,'#'#,*% -'#%# (+"'/ "%/ 3 1Q -'#%# (+"'/ 6+%#,%/. L% -%6+% /*$*'#-' '/)' '/-' %!%'%)*'#-"; ,%% !%' % #" ' // %+'+"/ !%% (")% '+ $'# ' +% '/,'#'#,*%.
S!"#')"/ &' '/ *$%+)'#-' !"6%6+' &' ,%% !%' ,"#-*63' ,"# ,%+&*'% ' +"/ "/ %+'+"/ 3 &'7 /* #" " "/ ' +"/ %+'+"/ '# # !% '/ ")*#%#-' R7 +% '/,'#'#,*% -'#8 (+"'/ "%/. C8+ '/ +% !"6%6*+*% ' &' # '/,'#*'#-' -'#$% % %+ )'#"/ # %+'+" ")*#%#-'7 6 %+ )'#"/ # %+'+" ','/*9"7 , # %+'+" ','/*9"7 %" &' +% '/,'#'#,*% -*'#' (+"'/ "%/ !
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
9'%7 3 /!"#$% &' ,%+&*' !% ' '8)'#'/ /"# *#'!'#*'#-'/. C8+ '/ +% !"6%6*+*% ' &' #% )8&*#% ,"#,+3% &' % ,%% #" ' -'/ ')!+'%"/ '/-8 )*#-*'#" ,%#" -""/ '/-8# *,*'#" +% 9'%7 6 %+ )'#"/ #" ' +"/ -'/ ')!+'%"/ '/-8 )*#-*'#" ,%#" -""/ '/-8# *,*'#" +% 9'% ! a) L%/ -'/ !'6%/ /"# *#'!'#*'#-'/. P" +" -%#-"7 +% !"6%6*+*% '/ ' ?7?
= ?.???12. b) P %+ )'#"/ # '" = 1 [ P /*# '"'/ = 1 [ ?. = ?.1. 2.? L% !"6%6*+*% '/')!'5 # !%!'+ '# +% )%#*!+%,*# ' +% +"-'% '+ '/-%" ' P'##/3+9%#*% '+ 2 ' %6*+ ' 1? L"/ Á#$'+'/ T*)'/7 ' /'!-*')6' ' 1?. P%% '-')*#% ,%% #" ' +"/ $*-"/ '+ #)'" $%#%" ' -'/ $*-"/7 ,%% #" ' +"/ #)'"/ ?7 17 27...7 /' ,"+",% '# #% !'+"-% ' !*#$‒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
( )=
50 !
50
H%3
3
?"
=
(
)
3 ! 50 −3 !
= 17?? )%#'%/ ' '+'$* +"/ $%#%"'/.
C%% #" ' '++"/ '/ *$%+)'#-' !"6%6+'.
()=
4!
4
a) H%3
3
"
=
(
)
3 ! 4 −3 !
= )%#'%/ '# &' +"/ "$%#*%"'/ $%#'#
-""/ +"/ !')*"/. L% !"6%6*+*% '/ Q1??.
b) H%3
( )( ) 4
46
2
1
(
=
4!
(
)
2 ! 4 −2 !
)(
46 !
(
)
1 ! 46−1 !
)=
2 )%#'%/ '# &' +"/
"$%#*%"'/ !''# $%#% "/ !')*"/ 3 #% ' +% "-%/ !'/"#%/ $%#' '+ -',' !')*". P" +" -%#-"7 +% !"6%6*+*% '/ 2Q1??.
( )( )
c) H%3
4
46
1
2
(
=
4!
( −1 ) !
1! 4
)(
46 !
( −2 ) !
2 ! 46
)=
1?. L% !"6%6*+*% '/
1?Q1??.
( )
46 !
46
d)
3
= " =
(
)
3 ! 46−3 !
= 171?. L% !"6%6*+*% '/ 11?Q1??
2.1 S* "/ '9'#-"/7 A 3 B7 /"# -%+'/ &' P A=.7 P B=.7 3 P A
( (
) ( ) = ( ) ) = ( ) ( ( )) =
P A ∩ B
.1
1
P A
.5
5
c) P AA ∪B =
P A
.5
P A ∪ B
.7
5
=
7
P A∪B = PA PB [ PA
(
P ( A ∩B ) P ( A ∩B )
e) P A
(
) = ) ) ) =
.1 .1
= 1
P ( A ∩ B
.1
P ( A ∪ B
.7
2.2 P%% ,*'-% !"6+%,*# ' ')!+'%"/7 '+ !",'#-%' &' %!'6% " '!'6% # '%)'# ' ,")!'-'#,*% '# # -%6%"7 *#*,%" ' %,'" ,"# '+ $4#'"7 ('"# ,")" /' 9' '# +% -%6+% /*$*'#-'. E/-" '/7 ' -"%/ +%/ !'/"#%/ &' -")%"# '+ '%)'#7 2\ '/-%6%# '# +% ,%-'$"% ' 0")6'[%!"6%"7 1\ '/-9*'"# '# +% ,%-'$"% ' )'['!"6%% 3 %/ /,'/*9%)'#-'. U# ')!+'%" 9% % /' /'+',,*"#%" %+ %% ' '/-% !"6+%,*#. S'% A '+ '9'#-" ' &' '+ ')!+'%" "6-'#$% #% ,%+*(*,%,*# ' %!"6%" '# '+ '%)'# 3 /'% M '+ '9'#-" ' &' /' /'+',,*"#' # 0")6'.
% L"/ '9'#-"/ A 3 M /"# *#'!'#*'#-'/ 6 L"/ '9'#-"/ Ᾱ 3 F /"# *#'!'#*'#-'/ ! 60
a) T'#$% '# ,'#-% &' P A =
100
= ?. 3 P A M = ?.2 Q ?. = ?.. P" +"
-%#-"7 A 3 M /"# *#'!'#*'#-'/. 40
b) S*)*+%)'#-'7 P Ᾱ =
100
= ?. 3 P Ᾱ F = ?.2 Q ?. = ?. =7 !" +" &' Ᾱ 3
F /"# *#'!'#*'#-'/. 2. G'$" M'#'+ (' # )"#' &'7 '# 17 /$** #% -'"% ' +% 0''#,*% 6%/%% '# +% ,*'#,*% ' +% $'#4-*,%. É+ *'#-*(*, *#*9*"/ 0'-'",*$"-"/ !%% ,"+" ' (+" &' -'#%# "/ %+'+"/ # = %+'+" ,"+" 6+%#," ','/*9" 3 # R= %+'+" ,"+" "" ")*#%#-'. C%#" '/-"/ *#*9*"/ /' %!%'%6%#7 /' "6/'9%6% &' Q ' +% '/,'#'#,*% -'#%# (+"'/ "%/ 3 1Q -'#%# (+"'/ 6+%#,%/. L% -%6+% /*$*'#-' '/)' '/-' %!%'%)*'#-"; ,%% !%' % #" ' // %+'+"/ !%% (")% '+ $'# ' +% '/,'#'#,*%.
S!"#')"/ &' '/ *$%+)'#-' !"6%6+' &' ,%% !%' ,"#-*63' ,"# ,%+&*'% ' +"/ "/ %+'+"/ 3 &'7 /* #" " "/ ' +"/ %+'+"/ '# # !% '/ ")*#%#-' R7 +% '/,'#'#,*% -'#8 (+"'/ "%/. C8+ '/ +% !"6%6*+*% ' &' # '/,'#*'#-' -'#$% % %+ )'#"/ # %+'+" ")*#%#-'7 6 %+ )'#"/ # %+'+" ','/*9"7 , # %+'+" ','/*9"7 %" &' +% '/,'#'#,*% -*'#' (+"'/ "%/ !
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a) L%/ -'/ !'6%/ /"# *#'!'#*'#-'/. P" +" -%#-"7 +% !"6%6*+*% '/ ' ?7?
= ?.???12. b) P %+ )'#"/ # '" = 1 [ P /*# '"'/ = 1 [ ?. = ?.1.
