TRABAJO COLABORATIVO 2 PROBABILIDAD
AMÉRICA LIDUEÑA MEZA COD. 32612763 DANILO ANDRES JOJOA ABELLA CÓDIGO: 107323766 SUSANA PAOLA AMARIS CÓDIGO: !27"3"7# .
T$%&' LUIS ALE(ANDER SARAVIA ROA
UNIVERSIDAD UNIVERSI DAD NACIONAL N ACIONAL ABIERTA ) A DISTANCIA DISTANCIA * UNAD ESCUELA DE CIENCIAS ADMINISTRATIVAS+ CONTABLES+ ECONÓMICAS ) NEGOCIOS ,ECACEN ADMINISTRACIÓN DE EMPRESAS EMPRESAS B&&%/+ N&4' # 5 201!
INTRODUCCIÓN. E % %'848 %'8489& 9& ;8 ;8% %8 8 2 %$ %$5 5& & 5 <8& <8& % % 5& 5& <$ <$%8 %8 ;8 %/%<8 5 ;8 $585 5& 5; =5$;& 5 '&484;585+ >$ %'8%8 &4' V8'84; V8'84; 8;8%&'8+ 5%'4$<= 5<'%8 ? <&%$8+ 558 5 %5<8 <%'8;+ ;8 5%'4$<= 4&8; %' &%'8 5%'4$<& 5 '&484;585. E%8'& &5& '/<%<8 %8 %/%<8 <& ; @ 5 58' &;$<= 8 ;& 5& <8& ;8%85&.
RESUMEN DE LOS CONCEPTOS TEÓRICOS DE LA UNIDAD 2 V8'84; V8'84; 8;8%&'8 5<'%8 ? <&%$8+ 8;&' '85& ? 8'88 V8'84; 8;8%&'8 5<'%8 5<'%8 $5 Variable Variable Aleatoria Discreta: S ;;88 V8'84; <&%8' $ <&9$%& 5 '$;%85& &4;. L8 V8'84; '84; A;8% A;8%&' &'8 8 D<' D<'%8 %8++ '' ''% %8 8 58%& 58%& <&%85 <&%85&+ &+ <&& <&& ; '& 5 8'%<$;& 5@<%$&& $8 $%'8 & ; '& 5 8<<5% 5 <8''%'8 &' 8& $ %85& 585&. C$85& $8 8'84 8'84; ; 8;8%& 8;8%&'8 '8 $5 $5 %&8' %&8' Variabl Variablee Aleatori Aleatoriaa Contina: Contina: C$85& 8;&' $8 <8;8 <&%$8 ; 5&8 V8'84; A;8%&'8 C&%$8. A $ $5& 5& ;& ;& & &4; 4; 8;& 8;&' ' 5 $8 $8 V8'84 '84; ; A;8% ;8%&' &'8 8 C&% C&% $8 $8 & & '<8% ;& 8;&' >$ <&% ; 8<& $%'8; <&%$&. L8 V8'84; '84; A;8%&' ;8%&'8 8 C&% C&%$8 $8 '' ''% %8 8 58%& 58%& 55& 55&+ + <&& <&& & & %&5& ;& &4; &+ 8;%$'8+ %'8%$'8+ 5%8<8 & '&5& 5 58. <%& %& @$5 @$58 8 %8 %8;; ; %$ %$5& 5& 5 ;8 ;8 Valor Valor Es!era"o Es!era"o:: $ <&< 5%'4$<& 5 '&484;585. D5 8< $<& 8& % <&<%& 8 5& 8;<85& 8;8% ; &<& 5 $'& ? ;& ;%& % 8& 8 5& 8;<85& &' &%'& '&@&8; >$ <8 ' %&8 5<& C&5<& 5 <'%5$4'. P8'8 &4%' ; 8;&' '85& 5 $8 8'84; 8;8%&'8 5<'%8+ $;%;<8& C858 8;&' >$ F%8 $5 8$' &' ;8 '&484;585 5 &<$''<8 5 V8;&' ? ;$& $8& ;& '&5$<%&. E $ '&5& &5'85& 5 ;& R$;%85& >$ '8 ; @$%$'&.
Varian#a: L8 558 & 8;&' '85& 5 $8 8'84; 8;8%&'8 5 <8; &'%8<8 %85%<8+ $ 5<'4 ; ;$8' 5&5 <%'8 ;8 5%'4$< 5%'4$<= = 5 '&484;585. '&484;585. N<%8& N<%8& <8'8<%'8' <8'8<%'8' ;8 8'84;5 8'84;585 85 ;8 5%'4$<=. Distribci$n bino%ial: E <858 '$48 5; '%& =;& & &4; 5& '$;%85&: ; $<& A HF%& ? $ <&%'8'& . L8 '&484;585 5; $<& $<& A <&%8%+ 5<'+ >$ & 8'8 5 $8 '$48 8 &%'8. S ''%8 &' . E; '$;% '$;%85& 85& &4% &4%5& 5& <858 <858 '$48 '$48 5 55 5% % 5 ;& '$;% '$;%85 85& & &4%5& 8%'&'%. L8 5%'4$<= 4&8; $; ''%8' &' BH+ . ; '& 5 '$48 5 >$ <&%8 ; '%&. ;8 '&484;585 5 F%&. L8 '&484;585 5 1 + ? ;8 ''%8& &' >. V8'84; V8'84; 8;8%&'8 4&8;: L8 8'84; 8;8%&'8 4&8;+ (+ '8 ; '& 5 F%& &4%5& <858 '$48 5; '%&. L8 8'84; 4&8; $8 8'84; 8;8%&'8 5<'%8+ =;& $5 %&8' ;& 8;&' 0+ 1+ 2+ 3+ +...+ $&5& >$ 8 '8;85& '$48. E9;&:
K 6+ 8; ;88' $8 &58 10 < ? &4%' 6 <8'8. 5%'4$<= = 4&8; 4&8; Distribci Distribci$n $n bino%ial bino%ial ne&ati' ne&ati'aa ( &eo%)t &eo%)tric rica: a: L8 5%'4$< 8% 8%8 8 & 5%'4 5%'4$< $<= = 5 P8<8 P8<8;; $8 '8 '8; ;8< 8<= = 5 ;8 5%'4 5%'4$< $<= = &F%'<8 5&5 ;8 8'84; 8;8%&'8 ( ; '& 5 8?& B'&$;; @<%$85& 8%8 >$ % ' F%&+ <& $8 '&484;585 <&%8% 5 F%& . S 5< %&< >$ ( % $8 5%'4$<= 4&8; 8%8 <& 8'/%'& ? ' 1+ 2+ 3+... @H++'
C'1
1
>' . '
'+'1+''2...
E ;8 5%'4$<= &F%'<8+ ;8 8'84; 8;8%&'8 %848 5@58 <&& ; '& 5 8?& B'&$;; <8'& 8'8 &4%' ; '' F%&. S$&8 8&'8 >$ 58 <&&<' ; '& 5 8?& 8%8 &4%' r F%& F%& % <8& ;8 8'84; 8;8%&'8 5&858 4&8; 8%8.
Distribci$n "e Poisson: E%8 &%'8 5%'4$<= 5 '&484;585 5<'%8 %; ;8 >$ ;8 8'84; 8;8%&'8 ''%8 ; '& 5 %& 55% >$ &<$'' 8 $8 ;&<585 <&%8%. L8 5%'4$<= 5 P&&+ ;;8858 8 &&' 8 S= D P&& '&484;%8 @'8$ @$ ; ''& 5<'4 5<'4';8 ';8++ ; '< '<8; 8; &5;& &5;& 5 '&484 '&484;5 ;585 85 ;8 ;85& 5& 8'8 8'8 88; 88;8' 8' '&4;8 5 ;8 5 '8+ <&@84;585 ? <&%'&; 5 <8;585 <&& ; '& 5 '&8 >$ ;;8 8 $ ;$8' 5%'85& $ %& 5@5&+ ;& 5@<%& 8 ;8' 8'8 ; 8%'8;+ ; '& 5 48<%'8 $ <$;%&+ ; '& 5 &; 8&%85& $ 8'%5& 5 @%4&;+ ; '& 5 @8;;8 5 $8 />$8 $8 &'8 & $ 58+ ;8 <8%585 5 <$;& >$ %'8%8 &' $8 8$%&%8+ ; '& 5 ;;8858 %;@=<8 &' $%&+ %<. C&& $5 &4'8' %'8%8 5 8;;8' ;8 '&484;585 5 &<$''<8 5 <$8;>$' '& &' $585 5 5<= H%&'8; & 8<8;. D85& $ %'8;& 5 '& '8;+ F% $5 55' $4%'8;& $@<%% >$&+ %8; >$: H1 H1 L8 '&4 '&484 84;;58 585 5 5 / / 5 $ 8< 8<'% '%& & $ $4 $4% %' '8 8;& ;& <'& <'& & @<8%. H2 L8 '&484;585 5 $8 &<$''<8 $ $4%'8;& ;8 8 8'8 %&5& ;& $4%'8;&+ ? '&&'<&8; 8 ;8 ;&%$5 5 %&. H3 E; <&%& 5 &<$''<8 <858 $4%'8;& 55% 5; 5 ;& 5/ $4%'8;&. E%&< ; '%& 8;8%&'& '<4 ; &4' 5 '&<& P&& & @;$9& 5 '&<& 5 P&&. U '&<& P&& <&%%$? $ <8& @<& 8;8%&'& ; <$8; ;& %& &<$'' 8; 88' $8 <8;8 5 %& H& 5 5%8<8. P&' 9;&+ ;8 &<$''<8 5 8<<5% $ <'$< <@<& 5 $8 <8''%'8 $ 5<& '&<&. C84 '<&'58' >$ & &4; '5<' <& 8<%%$5 ;8 <8%585 5 8<<5% >$ $5 &<$''' 5%'85& %'8;& 5 %&+ '& ; 8%'= 5 ;& 8<<5% '8 '& 5 5<& %'8;&.
Distribci$n *i!er&eo%)trica: E ;8 5%'4$<= 4&8; 8 >$ ; $ $%' %'& & 8< 8<8 8 <& <& ' '; ;8 8&+ &+ 8 8$' $'8 85& 5& ;8 5 5 5 5 < <8 8 5 ;& ;& 8?& ? ;8 '&484;585 <&%8%. S$=8 8&'8 >$ ; $%'& sin reemplazo+ <8& ; <$8; ;& 8?& & & 55%. S8 S8 N ; '& 5 ; ; %& %& 5 $ <&9$ 9$%& %& 5 ;& ;& <$8; $8; K & & 5%'85& <&& F%& ? NK <&& @8;;8+ %'8%8 8&'8 5 5%'8' ;8 '&484;585 5 F%& 8?& 5 ;& N ;%& 5; <&9$%& 5&5 ? . S8 %84F %84F ;8 8'84; 8'84; 8;8%&'8 8;8%&'8 ( ; '& '& 5 F%& F%& ;8 $%'8. E%&<+ ( % $8 5%'4$<= '&F%'<8 ? $ @$<= 5 5%'4$<= 5 '&484;585 %/ 5858 &': @H+N+K+ H KC . NKC NC 0+1+2+... HK+
Distribci$n ni+or%e "iscreta ( ni+or%e contin,a: L8 8'84; 8;8%&'8 5<'%8 / <;;8 8>$;;8 >$ %&8 =;& $ '& @%& 5 8;&' &4 &4; ; + <858 <858 $& $& <& ;8 8 '& '&484; 84;5 585 85.. E;;8 ;;8 5& 5& 8 8 %& %&< < 8' 8'84 84; ; 8;8% 8;8%&' &'8 8 5< 5<' '%8 %8 $@& $@&' ' ? $ 5% 5%' '4$ 4$< <= = $@ $@&' &' 5<'%8 %/ 5858 &': @H 1 P8'8 $8 8'84; 8;8%&'8 5<'%8 $@&' (+ >$ $5 %&8' ;& 8;&' 1+ 2+...+ + ;8 58 : ) $ 58<= %/58' : S 2 H2 112. S 5< >$ $8 8'84; X & & $8 5%'4$<= 5%'4$<= $@&' $@&' ; %'8;& %'8;& Qa+b+ $ $ @$ @$<= 5 5 5 5585 ;8 ;8 $%: C& %8 ;? 5 '&484;585+ ;8 '&484;585 5 >$ 8; 8<' $ '%& 8;8 8;8%& %&' '&+ &+ ; 8;& 8;&'' 5 X % % <& <&' '5 55 5& & <' <'%& %& $4 $4% %' '8 8;& ;& 5 Q a+b 55 <8% 5 ;8 ;&%$5 5; &+ & 5 $ &<=.
Distribci$n nor%al: E ; &5;& 5 5%'4$<= / $%;85& ;8 '/<%<8+ ?8 >$ $;%%$5 5 @=& <&&'%8 $8 5%'4$<= &'8;. E%8 5%'4$<= 5 <8'8<%'8 &'>$ ;& 8;&' 5%'4$? @&'85& $8 <888 5 G8$+ %&'& 8 $ 8;&' <%'8; >$ <&<5 <& ; 8;&' 5& 5 ;8 5%'4$<=. L8 distribución gaussiana + '<4 %84F ; &4' 5 distribución normal + ?8 >$ $8 '8 8?&'8 5 ;8 8'84; 8;8%&'8 <&%$8 5 ;8 8%$'8;8 $ %8 5%'4$<=. V8'H S2 : ; 8;&' 5& 5 ;8 5%'4$<= ? '<8% 5&5 %8 ; <%'& 5 ;8 <$'8 H5 ;8 <888 5 G8$. 2: ;8 8'88. I5<8 ;& 8;&' %/ / & & 8;985& 5; 8;&' <%'8;: ;8 8'88 4898 ;& 8;&' %/ '=& 8 ;8 58 8;%8+
%&< ;& 8;&' %/ $? 8;985& 5 ;;8. S ''%8 &' 2 &'>$ $ '8 <$85'858+ + ;8 5&858 58<= %/58' .
Distribci$n c*i ca"ra"o ( t "e st"en: D%'4$<= E %85%<8 %85%<8++ ;8 5%'4$<= H5 P8'&+ 5&5 '&$<8 <&& 9 <$85'85&+ $8 5%'4$<= 5 '&484;585 <&%$8 '&484;585 <&%$8 <& $ 8'/%'& k >$ >$ ''%8 ;& '85& 5 ;4'%85 5 ;4'%85 5 ;8 8'84; 8;8%&'8: D&5 Z i i & &'8;+ 5 58 58 <'& <'& ? 8'88 8'88 $&. $&. & 8'84; 5 5%'4$<= &'8;+ E%8 5%'4$<= '8 84%$8;% <&& L8 5%'4$<= 9<$85'85& % % $< $<8 8 8;< 8;<8< 8<& & @' @'<8 <8 %85 %85%< %<8 8+ &' 9;& ;8 5&858 '$48 $%;858 $%;858 <&& '$48 5 55<8 ? <&& '$48 5 4&585 5 89$% ? ;8 %8<= 5 8'88. T84F T84F %/ &;$<'858 ; '&4;8 5 %8' ;8 58 5 $8 &4;8<= &'8;% 5%'4$58 ? ; '&4;8 5 %8' ;8 5% 5 $8 '<%8 5 ''= ;8;+ ;8;+ 8 %'8F 5 $ 8; ;8 5%'4$<= % 5 S%$5%+ S%$5% + ? 8'%<8 %&5& ;& '&4; '&4;8 8 5 8/ 8/; ; 5 8' 8'8 88 8++ &' $ 8; ;8 5 5%'4 %'4$< $<= = 5 S5<&'+ >$ ;8 5%'4$<= 5; <&<% 5 5& 8'84; 8;8%&'8 5 5%'4$<= 9<$85'85& 55%. PROPIEDADES. $<= 5 5585 $<= 5 5%'4$<=
ESTUDIO DE CASO -
-.
La !rob !robab abil ili" i"a" a" "e e na na !ers !erson onaa e e lle& lle&e e al re+ re+&i &io o se seaa en'ia"o a 'er al "octor
(2023.!3.! PUNTUACION MEDIA23.! VARIBALE (3.! DESVIACION ESTANDAR7.! X μ σ −
Z
=
3.5 23.5 −
=
7.5
− =
20
7.5
=−
2.66
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ? 0.003" 0.03"W
2. La !robabil !robabili"a" i"a" "e e e na !erso !ersona na e lle&e lle&e al re+&i re+&io o ten&a na na !ntaci$n "e -/ o %enos !ntos x 10 μ 23.5 σ 7.5 =
=
Z
=
X μ σ −
=
=
10 23.5 −
7.5
=
−
13.5
7.5
1.8
=−
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ? 0.03!" 3.!"W
0. La !robabil !robabili"a" i"a" "e e e na !erso !ersona na e lle&e lle&e al re+&i re+&io o ten&a na na !ntaci$n entre -1 ( 2/ !ntos. P$%$8<= 16 x
=
Z
16 μ=23.5 σ =7.5
x μ σ
=
−
=
16
−
23.5
7.5
=
7.5
−
7.5
1
=−
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ? 0.1!#7 P$%$8<= 20 x 20 μ 23.5 σ 7.5 =
Z
=
=
X μ σ
=
−
20 23.5 −
=
7.5
− =
23.5
7.5
0.46
=−
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ?
0,3228
A&'8 $8 0.1!#7 0.1!#7 0,32280.#1! #.1!W
. Si las !erson !ersonas as sin *o&ar *o&ar con !nta !ntaci$ ci$n n en el -34 %5s %5s alto "eben "eben serr en'i se en'ia" a"as as a los los se ser' r'ic icio ioss "e !re' !re'en enci ci$n $n "e sic sici" i"io ios6 s6 78) 78)
!ntaci$n *ace cali+icar a na !ersona e lle&a al re+&io !ara este ser'icio9 23.!1!W27.02! Ó8 >$ ;8 $%$8<= >$ 8< <8;@<8' 8 $8 '&8 >$ ;;8 8; '@$& 5 '<& 5 '<= 5 $<5& 5 27.02! 85;8%.
3. Las Las !ers !erson onas as sin *o&ar *o&ar con !nta !ntaci ci$n $n en el 234 %5s bao6 bao6 se les les en';a a n ser'icio "e orientaci$n orientaci$n laboral !ara !ara %eorar ss recrsos. 78) !ntaci$n !er%ite cali+icar a na !ersona !ara acce"er a este ser'icio9 23.!2!W17.62! L8 $%$8<= >$ 8< <8;@<8' 8 $8 '&8 8'8 8<<5' 8; '<& 5 &'%8<= ;84&'8; 8'8 9&'8' $ '<$'& 5 17.62!.
ESTUDIO DE CASO 2.
-. Por %e"io %e"io "e las s!osic s!osiciones iones "e Seli&% Seli&%an6 an6 calcle calcle la !robabil !robabili"a" i"a" "e e la estatra "e n solo 'ar$n a"lto c*ino esco&i"o al a#ar sea %enor o i&al a -3 c% X
154 μ=166.3 σ =3.7
Z
X μ σ
=
=
−
=
154
−
166.3
3.7
=
−
12.3
3.7
=−
3.32
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ; 8;&' 0+000! 0.0!W
2. De i&al i&al %anera6 %anera6 cal calcl clee la !roba !robabil bili"a i"a" " "e e e la es estat tatra ra "e n solo 'ar$n a"lto c*ico esco&i"o al a#ar sea %a(or a -3< c% X
158 μ =166.3 σ =3.7
Z
X μ σ
=
−
=
158 166.3 −
=
3.7
− =
8.3
3.7
2.24
=−
V8& V8& 8 ;8 %84;8 5 5%'4$<= &'8; ; 8;&' 0+012! 1.2!W
0. Los reslta"os "e la !re&nta !robabili"a"es "e Seli&%an9
-
7concer"an
con
la s
N& <&<$'58 ?8 >$ ;8 5< >$ ; 2.!W %5'8 1! < 5 8;%$'8 & & ? &;& ; 0.0!W %5'8 8 8;%$'8.
. Co%e Co%ent ntee ac acer erca ca "e la 'a 'ali" li"e# e# "e las las s!o s!osi sicio cione ness "e Seli Seli&% &%an an 7=a( al&,n error b5sico en s ra#ona%iento9 S 8? ''&'+ S;8 55$9& ;8 %8%$'8 '&5& 5 ;& <& <&%8' <& ;& 58%& %85%<& '%%+ 5<' %&= ;8 %8%$'8 '&5& 5 ;& %85&$5. C& ;& 58%& %85%<& '%%+ 5<' ;8 5 ;& <&+ 54= <8;<$;8' ;8 58 $85& ; '&5$<%& 5 <858 8;&' 5 <& $ '&484;585 <&''&5%. P8'8 P8'8 <8;< <8;<$; $;8' 8' %8 %8 '&4 '&484 84;;5 585 85 S; S;8 8 85 85' '% % >$ >$ & % % ; >$8;% <& 5; S'<& 5 S8;$5 5 8 <&& E%85& U5& ? &' %8%&+ 5@<; &4%' ;8 %85%<8 5 8;$5 5 ;8 &4;8<= 8<%$8; <8. S 48'&+ 8@'8 >$ , '8; &% >$ ;8 ;&%$5 5 $ & 8; 8<' ''%8 ; 2#+6W 5 $ %8%$'8 @8;-. E%8 8@'8<= 54 %' $ 8' 5 ''&' $? 8;%& $%& >$ & $8 558 8<%8 & $8 $&<=.
3. Se&,n Se&,n criteri criterios os esta";s esta";stic ticos os se consi" consi"era era e n in"i'i" in"i'i"o o es "e estatra alta si s!era el !ro%e"io en %5s "e 2 "es'iaciones est5n"ar. est5n"ar. Usan"o las s!osiciones s!osiciones "e Seli&%an Seli&%an :
a> Calc Calcl lee "e e e la !rob !robab abil ili" i"a" a" "e n 'aron aron a"l a"lto to c*in c*ino o se seaa consi"era"o co%o "e estatra alta.
(
x μ σ
(
173,7
P ( x x ≥ 173,7 ) P z ≥ =
P ( x x ≥ 173,7 ) P z ≥ =
−
) −
166,3
3,7
)
P ( x x ≥ 173,7 ) P ( z ≥ 2 ) =
P ( x x ≥ 173,7 ) 0,0228 =
=
2,28
b> Calcle la !robabili li" "a" "e e n 'ar$n a"lto c*ino sea consi"era"o co%o "e estatra baa.
(
x μ σ
(
158,9
P ( x x ≤ 158,9 ) P z ≤ =
P ( x x ≤ 158,9 ) P z ≤ =
−
) −
166,3
3,7
)
P ( x x ≤ 158,9 ) P ( z z ≤ 2 ) =
−
P ( x x ≤ 158,9 ) 0,0228 =
=
2,28
1. Con base base en los reslta" reslta"os os anteriores anteriores66 ar&%ente ar&%ente si consi"e consi"era ra o no e Den& ?ia!in& to%o en centa la estatra al ele&ir a s scesor. N& <&5'& >$ D (8 $%;8'8 ;8 %8%$'8 <&& <'%'& ;8 ;<<= 5 $ $<&' $%& >$ ; W 5 ;8 &4;8<= %5 8 %' %8 %8%$'8 ;& $ $5 55$<' >$ <8 ;8 %85 5 ;8 &4;8<= $4'8 &55& ' <&58 &' % <'%'&.
CONCLUSIÓN
D 8<$'5& 8 ;8 %/%<8 5 ;8 $585 5& ? 8 ;& 9'<<& ;8%85& ;& 5& 5& <8& <8&+ + 5 5%@ %@< <8 8 ;& ;& <&< <&< %& %&+ + @$5 @$58 8 %& %& ? F%& F%&5& 5& 5 ;8 '&484;585+ %84F <&'5 ;& ;& '<& ? 8;<8<& 8;<8<& >$ % % ;8 '&484;585 ;& 5%%& <8&.
RE@ERENCIAS IILIOBRA@ICAS
M&'8; M&'8;+ + A5'88 5'88 H2010 H2010 M&5$; M&5$;& & P'&48 P'&484; 4;58 585. 5. B&&%/ B&&%/ D.C.+ D.C.+ U U' '585 585 N8<&8; A4'%8 ? 8 5%8<8 * UNAD. X8;&;+ X8;& ;+ R H1" H1""" "".. P'& P'&48 484; 4;58 585 5 ? % %85 85% %<8 <8 8' 8'8 8 '& '&. . Y8&8'<88.
MF<& MF <&
L;/++ Y. H20 L;/ H2006 06.. E% E%85 85% %<8 <8 5 5<' <'% %8 8 ? 5 5%'4 %'4$< $<& & 5 '& '&48 484; 4;58 585. 5. B&&%/: U&'%. U&'%.
VIDEOS CONSULTADOS. V8'84; V8'84; 8;8%&'8 5<'%8. R<$'85& 5: %%:.?&$%$4.<&8%< %%:. ?&$%$4.<&8%<[8E>DKI( [8E>DKI( D%'4$<& 5 '&484;585. R<$'85& 5: %%:.?&$%$4.<&8%<[$UZI6DM D%'4$<& <&%$8. R<$'85& 5: %%:.?&$%$4.<&8%< %%:. ?&$%$4.<&8%<[Z>6D< [Z>6D< %%:.?&$%$4.<&8%< %%:. ?&$%$4.<&8%<[MCY!K@M [MCY!K@M D%'4$<= B&8;. R<$'85& 5: %%:.?&$%$4.<&8%< %%:. ?&$%$4.<&8%<[P(UPI\ [P(UPI\
