fase2_grupo_100402_39

FASE FASE 2: DISTRIBUCIONES DIS TRIBUCIONES DE PROBABILIDAD

PRESENTADO POR: RUBY TATIANA CALIXTO RINCÓN CÓD: 1052020966 MARCO LEONARDO MORA CARLOS HERNANDO ANGEL CÓD: 111 1118550011 8550011 BAYRON ANDRES RUIZ AGUILERA COD: 111655008 CRISTIAN DANILO GARCIA ARDILA CÓD: CÓD: 111 11118555128 18555128

GRUPO: 10002!"9

#ORGE EDUARDO ROYERO TUTOR

UNI$ERSIDAD NACIONAL ABIERTA Y A DISTANCIA %UNAD& ESCUELA DE CIENCIAS B'SICAS TECNOLOG(A E INGENIER(A PROBABILIDAD CEAD YOPA YO PAL L 201)

INTRODUCCION E* +,-. . /,.**. ,3 ,+. 34,. .+77/./ , *. S*473 /,* ,+4/7 /, .  ;;4,+ 4+7*7<.3/ *. D7+7473 /, P.7*7/./, P .7*7/./, , 4. =4, ,* ,+4/7.3+, 34*+, , ,+, +,-. *,. .3.*7, > .;*7=4, * ;,/7-7,3+ ;.. ,33+. *. ,;4,+. ./, . 4 4

3 377-7, 7,3+ 3+  ./=47 ./=477 7/ / > /, ?-. ?-. *. *.. .+7 +7. . , @.. @..33 *. ,;, ,;,+ +7 7..

3*7/.73 /,* +.. /3/, =4,/,3 ;*.-./ +/ * .;+, @,@ S,3 *  ;.-,+ /, *. R7. > ,* S>**.4 L. F73. F73.*7 *7/. /.// /, ,+, ,+, +. +.. . , /,+, /,+,-73 73. . *. .;. .;.7 7/. /.// /, ./. ./. ,+4 ,+4/7 /7.3 .3+, +, ;.. ;.. /,,3 /,,3* *, ,, , ,3 43 . . ,;, ,;,? ?7 7 , , 43 .43+ .43+ ,3 ;.+7 ;.+74 4*. *.  ;.. ;.. .;*7 .;*7. . *. P.7*7/./ > 3+,+.3/ *. ;,43+. ,. ,* . ,7/ E* ,+7 , ,.*7<. /, M.3,. 73/77/4.* ,* /,.** /,* . E7/ > /.* . 3, ,3 ,* ? *..+7 ;.. =4, * /,- -;., @..3 4 ,;,+7. ,.*7-,3+.73 > @., *. ,;,+7. 3*7/.73 E* T.. C*..+7 , **, . . -,/7.3+, *. *473 /,* P*,-. ABC /3/, , .4+.3 . * ;.-,+ ,77/ ,3 *. 3-. APA , 7-;+.3+, .,. =4, , /,+,-73 /,+,-73 3 .3+,77/./ .3+,77/./ ,* * .73./ =47,3 , ,* ,3../ ,3../ /, 47 .* ,3+3 /, E.*4.73 > ,47-7,3+

DESARROLLO DEL TRABAJO 1. Cuadro sinóptico E* 4; /7,. > ;,,3+. ,3 43 4./ 73;+7 ,* ,4-,3 /, * 3,;+ +,7 /, *. 437/./ 2 =4, /.3 4+,3+ . *. *473 /, * ,+4/7 /, . ;;4,+ > *473./  ; ,* 4;

2.- Soución a !studio d! caso 1. Soucionado por" #arco L!onardo #ora ESTUDIO DE CASO 1 P.. ,3+,. 43. T.,+. /, /7+ ,3 * .3 /,* 4; A.* * .3.*7+. /,* .3 *.7?7.3  .*7?7.3 .* *7,3+, ,3 ?4373 /, *. ;.7*7/./ /, =4, ,4*+, ,3+.*, U3. +.*. @.7+4.* /, .*7?7.73, , *. 747,3+,:

L. .*7?7.73 , *. 4-. /, * ;43+ /, * ,7 4 P ,,-;* S4@7 BJ3 +7,3, -,3 /, 25 . %12 ;43+&K @. 77/ ,3 ,* -7- /-77*7 /4.3+, / . %0  ;43+&K /,/, @., 4.+ . , /4, /, 43 .4+-7* %1" ;43+& ; ,* =4, ,.*7<.  ;. /, )5 %6 ;43+&K ,.*7<. .+ /-+7 /, 200 %10 ;43+& > ;,, 43. 4,3+. /, @,=4, %" ;43+& L. .*7?7.73 =4, +,3/. ,. /,  D,;4 3 43. ,43/. +.*. , 37,+,3 *. .*7?7.73, ,3 ;.7*7/./, /, ,3+.7*7/./ /,* *7,3+, .:

L. ;43+4.73 /, S4@7 %& , +./47. ,3 43. ;.7*7/./ /, ,3+.7*7/./ .;7-./. /, 081 E3 +. ;.*.. 81 /, * *7,3+, - S4@7 ,3,..3 /73, . *. ;,.73, 3 +.,+. /,* .3 A 3+734.73 , -4,+.3 * ,4*+./ /, *. ,3+,7+. 3 * +, ;7*, *7,3+,

P,;., 43 73?-, ,3 ,* =4, - -37- 73*4>.: 1 C.*7?7=4, . ./. 43 /, ,+ *7,3+, > .*4*, *. ;.7*7/./ /, =4, ,4*+,3 ,3+.*, 2 C3 * /.+ .3+,7, 3+4>. 43. .7.*, .*,.+7. =4, ,;,,3+, ,* 3-, /, *7,3+, ,3+.*, ;.. ,* .3 > /,+,-73,: C4* , *. ;.7*7/./ /, =4, * +, ,4*+,3 ,3+.*, " C4* , *. ;.7*7/./ /, =4, 37343 ,. ,3+.*,  D,+,-73, *. /7+7473 /, ;.7*7/./ ++.* /,* 3-, /, *7,3+, ,3+.*, ,3+, ,+, 4; /, +, *7,3+, 5 R,/.+, 43 ,, ,4-,3 /, 4 @.**.<

1.- Cai$i%u! a cada uno d! !stos ci!nt!s & cacu! a pro'a'iidad d! %u! r!sut!n r!nta'!s S, +-.3 * ,4*+./ /, *. ,3+,7+. /, * T, ;7*, C*7,3+, > , *.7?7.3 ,3 *. T.*. @.7+4.* /, .*7?7.73,

No('r! d! ci!nt!

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Da*id Born puntos Ed/ard Br!ndan untos Ann #cLau+ in untos

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a+o Costo d! Cu!nta tota (!nsua *i*i!nda d! punto d! C!%u! s Auto(ó* so i aorros 10 "00 .-. 10 200

15 * @,=4, 10 " ;;7,+.  37343. 7 12

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6 095

7 096

Y. C37/ ,* ++.* /, ;43+ /, * ;7*, *7,3+, , ;,/, . .. *. ;.7*7/./ /, ,3+.7*7/./ - S, ;4,/, , ,3 *. +.*. 747,3+, ; ./. 43 /, * *7,3+,

No('r! d! os ci!nt!s Da*id Born Ed/ard Br!ndan Ann #cLau+in

tota puntos 6  07

ro'a'iidad r!nta'iidad 95 .73 81 .61 ) .5

2.- 8Cu9 !s a pro'a'iidad d! %u! os tr!s r!sut!n r!nta'!s:  ;r1 < r2 < r0= >  ;r1=?  ;r2=? ;r0=  ;r1 < r2 < r0= > 095  081 0) Q 0569"  100 Q 5)  P.7*7/./ /, ,3+.7*7/./: 5) 

0.- 8Cu9 !s a pro'a'iidad d! %u! nin+uno s!a r!nta'!: Ci!nt! 1 ;r1= Ci!nt! 2 ;r 2= Ci!nt! 0 ;r 0= p ;r1= > 1 - ;r1= > 1  095 Q 005 p ;r2= > 1 - ;r2= > 1  081Q 019 p ;r0= > 1 - ;r2= > 1  0)Q 026 or o %u!"  ;r=>  ;r1 < r2 < r0=> P %1&  P %2&  P %"&  ;r1= @  ;r2= @  ;r0= > 005  019  026Q 0002) L. P.7*7/./ /, =4, 37343 ,. ,3+.*, , /,* 0002)

. D!t!r(in! a distri'ución d! pro'a'iidad tota d! n(!ro d! ci!nt!s r!nta'!s !ntr! !st! +rupo d! tr!s ci!nt!s.  ;r1=>  ;r1< r2  ;r1=.  ;r2=. ;r0= > 005  019  026Q .25  ;r1= >  ;r1

 ;r2= >  %095  019  026&   %005  081  026&   %005  019  0)& >  ;r2= > 0069"  00105"  000)0"  ;r2= > .47  ;r1 < r2 < r0= >  ;r1= @  ;r2= @  ;r0= > 095  081 0)Q .3470  ;r2= >  ;r1  ;r2= >  %005  081  0)=   %095 019 0)&   %095 081026& Q  ;r2=> 00299)  01""5) 02000)  ;r2= > .04041 R!su(!n"  ;r= > 0002)  ;r1= > 0069  ;r2= > 0"6"61  ;r0= > 0569"  !!!!!!!!!!!!!!!!!! 

TOTAL

1

3. R!dact! un 'r!*! r!su(!n d! sus aa+os S, , ,3 ,* ,+4/7 /,* . =4, ,* *7,3+, D.7/ B3 , ,* *7,3+, 73/7./ ;.. *. +.3.73 T7,3, 43. 4,3. 7/. ,/7+77. > ,3 -;..73 3 * + *7,3+, ,3, * ,=477+ 3,,.7

0.- Soución a !studio d! caso 2. Soucionado por" Ru'& Tatiana Cai@to. ESTUDIO DE CASO 2 L. /7+7473 /, ;.7*7/./ 73-7.* , 43. /7+7473 /, ;.7*7/./ /7,+. =4, ,  ;,,3+. 3 -4@. ?,4,37. U3. /, 4 ..+,+7. 37+, ,3 =4, * @.> /  ;7*, ,4*+./ ,3 43 /,+,-73./ ,3.> /,* ,;,7-,3+ > * ,4*+./ 3 -4+4.-,3+, ,*4>,3+,K O+. ..+,+7. /, *. /7+7473 73-7.* , ,* @,@ /, =4, *. .7.*, .*,.+7. , ,* ,4*+./ /, 3+, E /,7 , 4,3+. ,* 3-, /, 7+ ,3 ,* 3-, ++.* /, ,3.> U3. +,,. ..+,+7. /, 43. /7+7473 73-7.* 37+, ,3 =4, *. ;.7*7/./ /, 7+ , *. -7-. /, 43 ,3.> . +

U3 ,+4/7 /,* D,;.+.-,3+ /, T.3;+, /, I**737 3*4> =4, )62 /, =47,3, 4;..3 * .7,3+ /,*.3+, /, * ,@4* 4+7*7<.. 73+43 /, ,47/./ E+ 737?7. =4, * / 4;.3+, /, *.  ;.+, /,*.3+,. 4+7*7<..3 73+43, /, ,47/./ S4;3. =4, /,7/, -;.. *. 73?-.73 3 ,* 4 .+4.* =4, , /. .* 73+43 /, ,47/./ ;.. * 4.* ,*,73. 43. -4,+. /, 10 ,@4*

P,,3+, 43 73?-, ,3 ,* =4, - -37- 73*4>.:

1 E+. 7+4.73 4-;*, 3 * 4;4,+ /, *. /7+7473 73-7.* I/,3+7?=4,* 2 E*., 43 /7..-. /, .. ;.. *. /7+7473 /, ;.7*7/./ 73-7.* =4, ,;,,3+. ,+. 7+4.73 " C4* , *. ;.7*7/./ /, =4, * 4;.3+, /, *. ;.+, /,*.3+,. ,3 ,.+.-,3+, 6 /, * 10 ,@4* ,*,73./ 4+7*7,3 73+43, /, ,47/./  C4* , *. ;.7*7/./ /, =4, * 4;.3+, /, *. ;.+, /,*.3+,. /, ; * -,3 6 /, * 10 ,@4* 4+7*7,3 73+43 /, ,47/./ 5 C4* , *. ;.7*7/./ /, =4, * 4;.3+, /, *. ;.+, /,*.3+,. /, -7- 6 /, * 10 ,@4* 4+7*7,3 73+43 /, ,47/./ 6 E34,3+, ,* .* ,;,./ /,* 3-, /, ,@4* ,3 * =4, * 4;.3+, /, *.  ;.+, /,*.3+,. 4+7*7<.3 ,* 73+43 /, ,47/./

1- 8Esta situación cu(p! con os supu!stos d! a distri'ución 'ino(ia: Id!nti$%u!os. •

S7 4-;*, 3 * 4;4,+ /, *. /7+7473 73-7.* >. =4, ,3 ,**. * ,7+,3 / ;7*, ,4*+./ /,3-73./ 7+ > ?.. =4, ,3 ,+, . ,* ,,3+ ,7+ , 4. 73+43 /, ,47/./ > ,* ?.. , 3 4.*:  ;Q *. ;,3. 4. ,* 73+43 /, ,47/./ /,3+ /, 43 ,@4* =Q *. ;,3. 3 4. ,* 73+43 /, ,47/./ /,3+ /, 43 ,@4*

•

L. ;.7*7/./ /, 7+ ,3 ,+, . , /,*

76.2 ( p =0.762)

 > ,+. , *.

-7-. ,3 +/ * . 73/,;,3/7,3+,-,3+, /, *. ;.7*7/./ /, 73+, , /,7 3 /,;,3/, /,* 3-, /, ,3.> =4, , /,,.3 37/,. •

L. .7.*, .*,.+7. ,3 ,+, +7; /, /7+7473 +-. .*, ,.*, /,/, , @.+. 43 3-, ?7 >. =4, , 43. /7+7473 /7,+. ,3 ,+, . ,* 3-, /, ,@4* ,3 * 4.*, 4 4;.3+, 4+7*7<.3 ,* 73+43 /, ,47/./ ;4,/, , 0 1 2 "  53

2- Ea'or! un dia+ra(a d! 'arras para a distri'ución d! pro'a'iidad 'ino(ia %u! r!pr!s!nta !sta situación n

12

 p

0)62

=

02"8

L. +.*. /, *. ;;7,/./, .*4*./. ,3 *. /7+7473 73-73.* 4.3/  +-. * .*, %0 1 2 "10&, *. 747,3+,:

)EFCUL ;@= O ""0"11E  08 126906E 1 06 22")2E 2 05 00002"89 0 6 0001)180)  00088011" 3 6 00"28)8" 4 2 0090218"2 5 6 01805"12 6 02568906 7 8 026)5 1 )

E3 /3/,  ?4,.*4*./. /,*.747,3+, -.3,.: n

∑= ( nk ) x

 P ( X = K ) =

k 

a

n − k 

k  0

( )

10 ° −8  P ( X =0 )= 10 ( 0,762 ) ( 0,238 ) =3,30∗10

0

( )

1 9 −6  P ( X =1 )= 10 ( 0,762 ) ( 0,238 ) =1,26∗10

1

( )

2 8 −5  P ( X =2 ) = 10 ( 0,762 ) ( 0,238 ) =2,23∗10

2

( )

3 7  P ( X =3 )= 10 ( 0,762 ) ( 0,238 ) =0,0002

3

( )

4 6  P ( X = 4 ) = 10 ( 0,762 ) ( 0,238 ) =0,001

4

( )

5 5  P ( X =5 ) = 10 ( 0,762 ) ( 0,238 ) =0,008

5

( )

6 4  P ( X =6 )= 10 ( 0,762 ) ( 0,238 ) =0,032

6

( )

7 3  P ( X =7 )= 10 ( 0,762 ) ( 0,238 ) =0,090

7

( )

8 2  P ( X =8 )= 10 ( 0,762 ) ( 0,238 ) =0,180

8

( )

9 1  P ( X =9 )= 10 ( 0,762 ) ( 0,238 ) =0,256

9

( )

10 0  P ( X =10 ) = 10 ( 0,762 ) ( 0,238 ) =0,246

10

E* /7..-. /, .. /3/, , -4,+. *. ?-. /, /7+7473 /, ;.7*7/./ 73-73.* , ,* 747,3+,:

0-

8Cu9 !s a pro'a'iidad d! %u! os ocupant!s d! a part! d!ant!ra !n !@acta(!nt! 4 d! os 1 *!cuos s!!ccionados utiic!n cinturon!s d! s!+uridad:

n =10

π =0.762 n− x

 P ( x )=C π  ( 1− π )  x

10 −6

 P ( x = 6 )=C ∗ 0.762 ( 1−0.762 ) 6

 P ( x = 6 )=792 ( 0.1057)(−5,999 )

 P ( x = 6 )=502.20

- 8Cu9 !s a pro'a'iidad d! %u! os ocupant!s d! a part! d!ant!ra d! por o (!nos 4 d! os 1 *!cuos utiic!n cinturón d! s!+uridad:

 P ( x ≥ 6 )= P ( x =6 ) + P ( x =7 ) + P ( x =8 )+ P ( x =9 )+ P ( x =10 )  P ( x ≥ 6 )=0,032 + 0,090 + 0,180 + 0,256 + 0,246

 P ( x ≥ 6 )=0.804

3- 8Cu9 !s a pro'a'iidad d! %u! os ocupant!s d! a part! d!ant!ra d! (9@i(o 4 d! os 1 *!cuos utiic!n cinturón d! s!+uridad:

 P ( x ≤ 6 )= P ( x =0 )+ P ( x =1 ) + P ( x =2 ) + P ( x =3 )+ P ( x = 4 )+ P ( x =5 ) + P ( x = 6 ) −8

−6

−5

 P ( x ≤ 7 )=3.3∗10 + 1.27∗10 + 2.2∗10 + 0.00024 + 0.0017 + 0.0088 + 0.0329  P ( x ≤ 6 )=0.1226

4- Encu!ntr! ! *aor !sp!rado d! n(!ro d! *!cuos !n os %u! os ocupant!s d! a part! d!ant!ra utiian ! cinturón d! s!+uridad: S,3 *. /7+7473 73-73.* ,* .* ,;,./ , /,?73, -:  E ( x )=n∗ p

 E ( x )=10∗ ( 0.762 )

 E ( x )=7,62

E* .* ,;,./ /,* 3-, /, ,@4* ,3 * =4, * 4;.3+, /, *. ;.+, /,*.3+,. 4+7*7<.3 ,* 73+43 /, ,47/./ , /, ) .;7-./.-,3+,

.- Soución a !studio d! caso 0. Soucionado por" Caros !rnando An+!. ESTUDIO DE CASO 0 E* B.*+ , *. *+,. ,3 *3,. /, C*-7. 4;,7./ ; ETESA %E-;,. +,7+7.* ;.. *. .*4/& E 43 4, =4, 37+, ,3 .,+. 6 5   " 3-, ,3 4.*=47, /,3 /, 43. -.+7< /,* 1 .* 5K E* 4./ ,.*. ,3 43 +.,+3 * 6 3-, =4, ,, L 3-, ,+3 ,;,,3+./ ,3 5 .*+. 34-,./. /,* 1 .* 5 C./. 3-, .;.,, 43. *. ,< > *. .*+. .3./. , ,*,73.3 73 ,,-;*.< E* ;,-7 .4-4*./ , ,3+,. . =47,3 @.. 737/7 * ,7 3-,

P,,3+, 43 73?-, ,3 ,* =4, - -37- 73*4>.:

1 E+. 7+4.73 4-;*, 3 * 4;4,+ /, *. /7+7473 H7;,,-+7. I/,3+7?=4,* %S4,,37.: D77/. * 5 3-, ,3 / 4;: .3./, > 3 .3./,& 2 U.3/ *. /7+7473 /, ;.7*7/./ H7;,,-+7. /,+,-73. *. ;.7*7/./ /, =4, ,* 4./ .7,+, * 6 3-, " L. ,-;,. ,3../. /,* +, 73?-. =4, *. ;.7*7/./ /, =4, 737/.3 +/ * 3-, , /, 1 ,3 815060 4 737?7. ,+ ,3 +-73 /, ;.7*7/./ C737/, ,+ 3 4 ,;4,+. .3+,7

E* +, +.-73 +. 43 ;,-7 7 ,* 4./ @., 737/7 "   5 /, * 3-, .3./,

& C.*4*, *. ;.7*7/./ ;.. @., 737/7 " /, * 6 3-, .3./, 5& C.*4*, *. ;.7*7/./ /, =4, 737/.3  /, * 6 3-, .3./, 6& C.*4*, *. ;.7*7/./ /, =4, 737/.3 5 /, * 6 3-, .3./, )& C3 ., ,3 * ,4*+./ +,37/ 4+,/ 73,+7. /73, ,3 ,* BALOTO

1.- 8Esta situación cu(p! con os supu!stos d! a distri'ución ip!r+!o(Gtrica: Id!nti$%u!os ;Su+!r!ncia" Di*ida os 3 n(!ros !n dos +rupos" +anador!s & no +anador!s= E+. 7+4.73 4-;*, 3 * 4;4,+ /, *. /7+7473 @7;,-+7. /,7/ . =4, ./. .7.*, .*,.+7. +7,3, / ;7*, ,4*+./ 7+  ?.. ,3 ,+, . ,3 *. ,*,73 /, *. .*+. ./,- 4-;*, 3 =4, , 4. +,3, ,* 3-, /, 7+ ,* +.-. /, -4,+. , -.> . 43 > ?73.*-,3+, * 3 ,*,-,3+ /, *. -4,+. .*,.+7. 3 /,;,3/7,3+, /,7/ . =4, *. ,+.73 /, 43. .*+. 73?*4>, ,3 ,* ,4*+./ ?73.* /,*  .*+ +,37,3/ ,3 4,3+. =4, 3 73 ,,-;*.< S4 ?4373 /, ;.7*7/./ ,:

− ( )( − ) ( )= () m  N  m k  n k 

 P  x

 N  n

D3/,

() a b

=

a! b ! ( a −b ) !

D3/, N ,;,,3+. ,* 3-, ++.* /, *. ;*.73 3 ,* 34-, +-./ /, *. -4,+. ,* 3-, /, 7+ ,7+,3+, ,3 *. ;*.73 > V ,* 3-, /, ,*,-,3+ ,3 *. .+,. /, 7+ =4, , /,,.3 ,+4/7.

2.- Usando a distri'ución d! pro'a'iidad ip!r+!o(Gtrica d!t!r(inar a pro'a'iidad d! %u! ! Hu+ador aci!rt! os 4 n(!ros E3 ,+, .

− ( )( − ) = )= ( )

 P ( X   x

6 45 6 k  6 k  45 6

− ( )( − ) = )= ( )

 P ( X  6

 P ( X =6 )=

 P ( X =6 )=

6 6

45 6 6 6 45 6

( )( ) ( ) 6 39 6 0 45 6

( 1 ) (1 ) ( 8145060 )

 P ( X =6 )=0.00000012

L. ;.7*7/./ /, =4, 43 4./ +,3. 6 .7,+ , /,* 0000012

0.- La !(pr!sa !ncar+ada d! sort!o in$or(a %u! a pro'a'iidad d! %u! coincidan todos os n(!ros !s d! 1 !n 6134. 8uG si+ni$ica !sto !n tGr(inos d! pro'a'iidad: E+ 737?7. =4, ,7+,3 815060 ;7*, -73.73, /, 6 3-, ,3 43 ++.* /, 5 /3/, * 1 -73.73 , *. ,+. > .3./. C737/, ,+ 3 4 ,;4,+. .3+,7 E* +, +.-73 +. 43 ;,-7 7 ,* 4./ @., 737/7 "   5 /, * 3-, .3./,

= Cacu! a pro'a'iidad para ac!r coincidir 0 d! os 4 n(!ros +anador!s. − ( )( − ) = )= ( )

 P ( X   x

 P ( X =3 )=

6 45 6 k  6 k  45 6

( )( ) ( ) 6 3

45−6 6−3 45 6

( )( ) = )= ( )

 P ( X  3

 P ( X =3 )=

6 39 3 3 45 6

( 20 ) ( 9139 ) ( 8145060 )

 P ( X =3 )=0.0224

L. ;.7*7/./ /, =4, 43 4./ 737/. ,3 " 3-, , /, 22

3= Cacu! a pro'a'iidad d! %u! coincidan  d! os 4 n(!ros +anador!s − ( )( − ) = )= ( )

 P ( X   x

 P ( X = 4 ) =

6 45 6 k  6 k  45 6

( )( ) ( ) 6 4

45 −6 6− 4 45 6

( )( ) = )= ( )

 P ( X  4

 P ( X = 4 ) =

6 4

39 2

45 6

 ( 15 ) ( 741 ) ( 8145060 )

 P ( X = 4 ) =0.00136

L. ;.7*7/./ /, =4, 43 4./ 737/. ,3  3-, , /,* 01"6

4= Cacu! a pro'a'iidad d! %u! coincidan 3 d! os 4 n(!ros +anador!s − ( )( − ) = )= ( )

 P ( X   x

6 45 6 k  6 k  45 6

− ( )( − ) = )= ( )

 P ( X  5

 P ( X =5 ) =

 P ( X =5 ) =

6 5

45 6 6 5 45 6

( )( ) ( ) 6 39 5 1 45 6

( 6 ) ( 39 ) ( 8145060 )

 P ( X =5 ) = 0.00002873

L. ;.7*7/./ /, =4, 43 4./ 737/. ,3 5 3-, , /,* 00028)"

5= Con 'as! !n os r!sutados o't!nidos 8Ust!d in*!rtira din!ro !n ! BALOTO:  N 73,+7. ,3 ,* .*+ /,7/ . =4, *. ;.7*7/./, /, .3. 3 -4> .. 73*4 /7-734>,3/ ,* 3-, /, .7,+ *. ;.7*7/./, /, .3. ,3 4.*=47, -/.*7/./ "  5 .7,+ , -4> .. /,7/ . =4, *. .3+7/./ /, ++.* /, ;77*7/./, , .*+.

3.- Soución a !studio d! caso . Soucionado por" Ba&ron Andr!s Rui. ESTUDIO DE CASO  L. /7+7473 /, ;.7*7/./ /, P73 /,7, ,* 3-, /, ,, =4, , ;,,3+. 43 ,,3+ /4.3+, 43 73+,.* ,;,?7 E* 73+,.*  ;4,/, , /, +7,-; /7+.37. ,.  *4-,3 L. /7+7473 , .. ,3 / 4;4,+ E* ;7-, 37+, ,3 =4, *. ;.7*7/./ ,  ;;73.* . *. *37+4/ /,* 73+,.* E* ,43/ 4;4,+ 37+, ,3 =4, * 73+,.* 3 73/,;,3/7,3+, E3 +. ;.*.. 4.3+ - .3/, ,. ,* 73+,.* -.> , *. ;.7*7/./K ./,- ,* 3-, /, ,, =4, , ;,,3+. 43 ,,3+ ,3 43 73+,.* 3 73?*4>, ,3 * /,- 73+,.* E+. /7+7473 ;,, /7,. .;*7.73, S, *, 4+7*7<. - -/,* ;.. /,77 *. /7+7473 /, ,, ,3 43. ,3+./. /, /.+ ,* 3-, /, .>3, > +. 7-;,?,73, ,3 *. .73. /, .4+-7*, ,73 ;73+./ ,* 3-, /, ;.+, /,?,+4. ,3 ,3 ,* 3-, /, *7,3+, =4, ,;,.3 -,. ,3 43 ,+.4.3+, ,* 3-, /, .7/,3+, ,3 43. .,+,. ,3 43 ;,7/ /,+,-73./

L. -;.. /, .7.73 D,*+. A7*73, , ..+,7<. ; 4 ,;3.7*7/./ > 47/./ 3 ,* ,=47;., /, 4 ;.., ; * =4, ;. ,, , ;7,/, ,=47;., E3 *. -.>. /, * 4,* 3 , ;7,/,3 -.*,+.K ,3 .*43 , ;7,/, 43.K ,3 43 4.3+ , ;7,/,3 /K ;. ,, , ;7,/,3 +, ,+ S4;3. =4, 43. -4,+. .*,.+7. /, 1 000 4,* .. 43 ++.* /, "00 -.*,+. ;,/7/. D, ,+. -.3,. ,* 3-, ;-,/7 /, -.*,+.  ;,/7/. ; 4,* , /, 0" P,;., 43 73?-, ,3 ,* =4, - -37- 73*4>.:

1 E+. 7+4.73 4-;*, 3 * 4;4,+ /, *. /7+7473 P73 I/,3+7?=4,* 2 D,+,-73, 4* , *. ;.7*7/./ /, =4, ,3 43 4,* 3 , ;7,/. 37343. -.*,+. " D,+,-73, 4* , *. ;.7*7/./ /, =4, ,3 43 4,* , ;7,/. ,.+.-,3+, 43. -.*,+.  D,+,-73, 4* , *. ;.7*7/./ /, =4, ,3 43 4,* , ;7,/.3 ,3+, / > 4.+ -.*,+. 5 P/. ,+.*,, 4* , *. ;.7*7/./ /, =4, , ;7,/.3 ,3 43 4,* - /, 4.+ -.*,+. 6 E3 =4 --,3+ /,, ;,@. ,* 4;,7 /, *. A,*3,. =4, ,3 43 4,* , ,+3  ;,/7,3/ /,-.7./. -.*,+. P,;., 43 73?-, ,3 ,* =4, - -37- 73*4>.:

1- 8Esta situación cu(p! con os supu!stos d! a distri'ución oisson: Id!nti$%u!os C4-;*, 3 ,* ;7-, 4;4,+ =4, /7, =4, *. ;.7*7/./ , ;;73.* . *. *37+4/ /,* 73+,.* ,3 ,+, . /, 1000 4,* , ;7,/,3 "00 -.*,+. , /,7 =4, ,7+, 43  ;-,/7 /, 0" -.*,+. ;,/7/. ; 4,* S7 4-;*, 3 ,* ,43/ 4;4,+ /,7/ . =4, ,* 3-, /, ,, =4, , ;,,3+. 43 ,,3+ ,3 43 73+,.* 7 73?*4>, ,3 * /,- 73+,.* >. =4, , 43. -4,+. .*,.+7. > * 73+,.* 3 73/,;,3/7,3+, , /,7 7 , - .3/, ,* 73+,.* +,3/ -.>   ;.7*7/./ 2- D!t!r(in! cu9 !s a pro'a'iidad d! %u! !n un *u!o no s! pi!rda nin+una #a!ta n

 λ =

 x ∑ =

i

i 1

n  x

− λ

 λ . e f  ( x ; λ )=  x !

,: , *. ., /,* *.7+- 3.+4.* 74.* . 2)1828

V , ,* 3-, /, 4,37. /, 43 ,,3+K *. ;.7*7/./ /, =4, 7,3, /./. ;  *. ?4373 VW , *. ?.+7.* /, V   , 43 3-, ,.* ;7+7 74.* =4, ,* 3-, ,;,./ /, 4,37. /4.3+, ,* 73+,.* /./ L ,,3+ 4,3 ,3 ;-,/7 "00 -.*,+. ;/7/. ;  1000 4,* 0

−0.3

0,3 ¿ . (e

¿ ¿  P ( 0 )=¿

)

T,37,3/ ,3 4,3+. * .3+,7 ;/,- /,7 =4, @.> 43 ) /, ;.7*7/./ /, =4, 37343. -.*,+. , ;7,/.

0- D!t!r(in! cu9 !s a pro'a'iidad d! %u! !n un *u!o s! pi!rda !@acta(!nt! una (a!ta

1

−0.3

0,3 ¿ . (e

¿ ¿  P (1 )=¿

)

E3 ., .* ,4*+./ +,37/ , ;4,/, /,7 =4, *. ;.7*7/./ /, =4, , ;7,/. * 43. -.*,+. , /,* 222

. D!t!r(in! cu9 !s a pro'a'iidad d! %u! !n un *u!o s! pi!rdan !ntr! dos &cuatro (a!tas. 2

−0.3

0,3 ¿ . (e

¿ ¿  P (2 )=¿ 3

−0.3

0,3 ¿ . (e

¿ ¿  P (3 )=¿

)

)

4

−0.3

0,3 ¿ . ( e

¿ ¿  P ( 4 ) =¿

)

D, .4,/ . * ,4*+./ +,37/ , ;4,/, /,7 =4, ,* """  ;7,/, / -.*,+. =4, ,* 0" ;7,/, +, -.*,+. ,* 0025 ;7,/, 4.+ -.*,+.

3. odra !sta'!c!r cu9 !s a pro'a'iidad d! %u! s! pi!rdan !n un *u!o (9s d! cuatro (a!tas L. /7+7473 /, P73 3 ;,-7+, 3, *. ;.7*7/./ ,3 4 73+,.* 37/ > ,* /,+,-73. - /,  -.*,+. .* 3 ?7., 43 73+,.* , 7-;7*, @.,* ; -,/7 /, ,+, 7+,-. /, /7+7473 /, ;.7*7/./ 4. 8En %uG (o(!nto d!'! sosp!car ! sup!r*isor d! a A!ron!a %u! !n un )u!o s! !st9n p!rdi!ndo d!(asiadas (a!tas: L. ;.7*7/./ /, =4, ,3 43 4,* , ;7,/.3 +, -.*,+. , 74.* .* 0""  > .* .4-,3+. ,* 3-, /, -.*,+. *. ;.7*7/./ . .+,3/, ./. ,< - . , ; +.* .<3 ,3 ,* --,3+ =4, , ;7,/.3 - /, +, -.*,+. ,* 4;,7 /, *. .,*3,. /,, ;,@. =4, ,3 ,* 4,* , ,+3 ;,/7,3/ /,-.7./. -.*,+. > .* ,+ -.* ,3 ,* ;,

4.- Soución a !studio d! caso 3. Soucionado por" Cristian Danio Karcia. ESTUDIO DE CASO 3 P.. 43. ;*.73 .3/, /, ;,3. 73 @. 3 > P7*7.73 %2001& ,.-73.3 ?.+, /, ,+ ,4 > .+.-7,3+ ;7*7 ,-;*,.3/ *. E.*. /, D,;,73 /,* C,3+ /, E+4/7 E;7/,-7*7 %CESD& 43 4,+73.7 /, ,.*4.73 -437+.7

E3+, *. ;,3. 73 @. *. ;43+4.73 -,/7. /,* 4,+73.7 CESD , 225 3 43. /,7.73 ,+3/. /, 85 > , 37/,. =4, ;.. *. $.7.*, X Q ;43+4.73 /,* CESD a distri'ución !s nor(a. C- +.../ ,3 ,* ,. /, ./-773, ,3 43 ,?47 ;.. ;,3. 73 @. 4+,/ , ,* ,3../ /, .;*7. ,* CESD > /,, ,.*4. * ,4*+./ ;.. *. 34,. ;,3. =4, **,4,3 .* ,3+

D,3+ /, *. ;*+7. /,* ,?47 , ,34,3+. =4, 4.*=47, ;,3. 4>. ;43+4.73 ,. /, 21  - ;43+ ,3 ,* CESD /,, ,37., . , . 43 /+

P,;., 43 73?-, ,3 ,* =4, - -37- 73*4>.:

1 L. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 ,. ,37./ . , .* /+  2 L. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 +,3. 43. ;43+.73 /, 10  -,3 ;43+ " L. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 +,3. 43. ;43+4.73 ,3+, 16 > 20 ;43+  S7 *. ;,3. 73 @. 3 ;43+4.73 ,3 ,* 15 - .*+ /,,3 , ,37./. . * ,77 /, ;,,373 /, 477/7 4 ;43+4.73 @., .*7?7. . 43. ;,3. =4, **,. .* ,?47 ;.. ,+, ,77 5 L. ;,3. 73 @. 3 ;43+.73 ,3 ,* 25 - . , *, ,3. . 43 ,77 /, 7,3+.73 *..* ;.. -,. 4 ,4 4 ;43+4.73 ;,-7+, .*7?7. . 43.  ;,3. ;.. .,/, . ,+, ,77

1- La pro'a'iidad d! %u! una p!rsona %u! !+u! a r!$u+io s!a !n*iado a *!r a doctor. ¿ D.+ /,* ,,77 /3/,  P ¿ =

Ƶ 

=

Ƶ 

 X − µ δ  21 −22.5 8.5

=−0.17

Ƶ 

 X ≥ 20 ¿

E+, .* * 7/,3+7?7.- ,3 *. +.*. /, /7+7473 3-.*:

)ALOR  01)

AREA BAJO LA CUR)A 006)5

¿ P.. .*4*. *. ;.7*7/./ *77+./.  P ¿

 X ≥ 20 ¿   ,3+3,:

',. -;,3/7/. ,3+, Z .*4*.3/ > *. /,,@. /, *. .-;.3. .: 006)5  05 Q 056)5

RTA" *. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 > ,. ,37./ . , .* /+  , /, 56)5M 2- La pro'a'iidad d! %u! una p!rsona %u! !+u! a r!$u+io t!n+a una puntación d! 1 o (!nos puntos. ¿ D.+ /,* ,,77 /3/,  P ¿

=

Ƶ 

=

Ƶ 

 X ≥ 20 ¿

 X − µ δ  10 −22.5 8.5

=−1.4

Ƶ 

E+, .* * 7/,3+7?7.- ,3 *. +.*. /, /7+7473 3-.*:

)ALOR  1

AREA BAJO LA CUR)A 0192

¿ P.. .*4*. *. ;.7*7/./ *77+./.  P ¿

 X ≥ 20 ¿

  ,3+3,:

',. -;,3/7/. ,3+, Z .*4*.3/ > *. /,,@. /, *. .-;.3. .: 05  0192 Q 00808

RTA" *. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 > +,3. 43. ;43+4.73 /, 10  -,3 ;43+ , /, 808 0- La pro'a'iidad d! %u! una p!rsona %u! !+u! a r!$u+io t!n+a una puntuación !ntr! 14 & 2 puntos. S, +7,3, ,3 * /.+ /,* ,,77 =4,  P [ X ≤ 20 ; y X ≥ 16 ]

 X −µ  X −µ Ƶ 2= δ  δ 

Ƶ 1

=

Ƶ 1

=

Ƶ 1

=−0.76 Ƶ 2=−0.29

16 −22,5 8,5

Ƶ 2

=

20 −22,5 8,5

I/,3+7?7.3/ ,+, .* /, <,+. ,3 *. +.*. /, /7+7473 3-.* , +7,3, =4,:

)ALOR  0)6 029

AREA BAJO LA CUR)A 02)6 0111

P.. .*4*. *. ;.7*7/./ *77+./. =4, P  X ≤ 20 ; y X ≥ 16  E3+3,: ',. -;,3/7/. ,3+, Z1 > Z2 .*4*./ . *. I<=47,/. /, *. .-;.3. ; , .*, /, Z 3,.+7 .:  P [ X ≤ 16 ; y X ≥ 20 ]= Ƶ 1−Ƶ 2  P [ X ≤ 16 ; y X≥ 20 ]= 0,2764− 0,1141= 0.1623

RTA" *. ;.7*7/./ /, =4, 43. ;,3. =4, **,4, .* ,?47 > +,3. 43. ;43+4.73 ,3+, 16 > 20 ;43+ , /, 162" - Si as p!rsonas sin o+ar con puntuación !n ! 13M (9s ato d!'!n s!r !n*iadas a os s!r*icios d! pr!*!nción d! suicidios 8uG puntuación ac! cai$icar a una p!rsona %u! !+a a r!$u+io para !st! s!r*icio:

E3 ,+, ,,77 , ;,+,3/, . ;.+7 /, *. ;.7*7/./  ;,3+., /./ **,. .* 3-,  ;43+4.73 X P.. ,** /,;,.- X ,3 *. ?-4*.

=

Ƶ 

 X − µ  /3/, δ 

 X =( Ƶ δ )+ µ

T,37,3/ ,3 4,3+. =4, 15 Q 01500 ,3 *. +.*. /, /7+7473 ,3+3, , ,34,3+. Z  ;7+7 ; 7 . *. /,,@. /, *. .-;.3.

)ALOR  0"8

AREA BAJO LA CUR)A 01500

A@. 3 ,* .* /, Z ,-;*.<.- ,* .* ,3 *. ,4.73 /,;,./. .3+,7-,3+, ;.. @.**. X:  X =( Ƶ δ )+ µ= ( 0,38∗ 8,5 )+ 22,5=25.73

RTA" L. ;43+4.73 /, 43. ;,3. =4, **,. .* ,?47 ;.. =4, ,. ,37./. .* ,77 /,  ;,,373 /, 477/7 , 43. .*7?7.73 /, 25)"  - 3- Las p!rsonas sin o+ar con puntación !n ! 23M (9s 'aHo s! !s !n*a a un s!r*icio d! ori!ntación a'ora para (!Horar sus r!cursos. 8uG puntuación p!r(it! cai$icar a una p!rsona para acc!d!r a !st! s!r*icio: E3 ,+, ,,77 .* 74.* =4, ,* .3+,7 , ;,+,3/, . ;.+7 /, *. ;.7*7/./  ;,3+., /./ **,. .* 3-,  ;43+4.73 X P.. ,** /,;,.- X ,3 *. ?-4*.

=

Ƶ 

 X − µ  /3/, δ 

 X =( Ƶ δ )+ µ

T,37,3/ ,3 4,3+. =4, 25 Q 02500 ,3 *. +.*. /, /7+7473 ,3+3, , ,34,3+. Z 3,.+7 ; 7 . *. 7<=47,/. /, *. .-;.3.

)ALOR  06)

AREA BAJO LA CUR)A 02500

A@. 3 ,* .* /, Z ,-;*.<.- ,* .* ,3 *. ,4.73 /,;,./. .3+,7-,3+, ;.. @.**. X:

 X =( Ƶ δ )+ µ= (−0,67∗8,5 )+ 22,5 =16.80

RTA" L. ;43+4.73 /, 43. ;,3. =4, **,. .* ,?47 ;.. =4, ,. ,37./. .* ,77 /, 7,3+.73 *..* ;.. -,. 4 ,4 , 43. .*7?7.73 /, 1680  -,3

REERENCIAS BIBLIOKRICAS

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