Unidad 16 - Distribuciones de Probabilidad, problemas resueltos

8/10/2019 Unidad 16 - Distribuciones de Probabilidad, problemas resueltos

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1 Bachillerato - Matemticas - Unidad 16 Distribuciones de probabilidad

Pg.3 ! I Aqu queda manifiesta la estupidez de una presuncin docente Sera ms instructivo que fuesen 500 los lanzamientos, ocupando mayor tiempo del discente!ara resta"lecer el sentido com#n, asumiremoscinco (5) lanzamientos" # $ %r&ico de resultados Diagrama de barras1 ' 1 ( ' 1 ' ( 3 ) 3 () 6 ) ( * ' * (

6 ( Pg.3 ! II $!or qu% no e&tender los lanzamientos a '(000, para mayor ocio del discente)

!ara reconducir el escassimo sentido com#n docente, asumimosdiez (10) lanzamientos 3

" 1 ' 3 ) * 6 + ! 1, 2 $ 3 1 6 * ' 1 3 ) * 3 1

1 2 3 4 5 6

Pg.3+, 1 leatoria discreta /erminaci0n del up0n de la 2" 4,515'535)5*565 5+5!ali&icaci0n clsica de e7amen 4,515'535)5*565 5+5!51,

Pg.3+, 2 leatoria continua 8ongitud de los lpices de una clase 4 de 3 a 1+ cm /ra9ectos de alumnos de casa a colegio 4 de *, a *,,, m

Pg.3+1 3 a /abla de di&erencias1 2 3 4 5 6 1 , 1 ' 3 ) *

2 1 , 1 ' 3 ) 3 ' 1 , 1 ' 3 4 3 ' 1 , 1 ' 5 ) 3 ' 1 , 1 6 * ) 3 ' 1 , :unci0n de probabilidad; & i * 0 1 2 3 4 5

pi * 6/36 10/36 8/36 6/36 4/36 2/36 '+ 5+'- .+ '+/ '+ '+'-

b & i pi & i pi & i . pi 0 '+/ 0 0 Media; == 1835 94,1 ' 5+'- 5+'- 5+'- . .+ + -+ iaci0n t?pica; == 07,2 44,1 5 '+'- 5+'- .5+'- 1 35/18 35/6

c ( ) ( ) ==+++=+++= 18151834536192185613 x P 65

83,0

1@'1


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Pg.3+3 4 efectuososTornillosD N x Distribuci0n binomial parmetros; 3= x 30,0 p

a ( ) ==

== 027,013,03

33 3 x P 027,0

b ( ) ( ) == == 7,009,033,013,0

2

32 2 x P 189,0

c ( ) ( ) == == 343,017,01

0

30 x P 343,0

Pg.3+3 5 l nAmero de indi>iduos 416 es bastante modestolegido uno 41 5 la selecci0n del ' no se hace en las mismas condiciones ue el 18a >ariablen2 de varones seleccionados no sigue una distribucin binomial8as probabilidades ha9 ue calcularlas por mCtodos ms elementalesE

a ( ) ===== 141516391066

23,16

2,10

C

C

lesCasosPosibablesCasosFavor

x P 56

27 482,0

b ( ) ( ) ====lesCasosPosib

ablesCasosFavor x P x P 1011 ==

141516456

113,16

3,6

C

C

28

27

964,0

Pg.3+3 6 Fe trata de una distribuci0n binomial5 de parmetros ( )30,010 BeatoriaVariableAl x = 4n personas &a>orables a la gesti0n

a ( ) ( ) === 733103 7,03,023

891030,0130,0

3

103 x P 267,0

'@'1


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b ( ) =

== 10103,013,010

1010 x P 0000059,0

Pg.3+3 7 asistentes x = ( Distribuci0n binomial; ( )4,010 B

a ( ) ( ) === 733103 6,04,023

89104,014,0

3

103 x P 215,0

b ( ) ( ) ( ) ( ) ( ) ( )[ ] ==+=+=+===> 32101313 x P x P x P x P x P x P

= x x x 10

3

0

6,04,010

1 [ ] =+++ 215,0121,0040,0006,01 618,0

Pg.3+) 8 s una distribuci0n binomial ( )75,08 B" medio de descendientes con lana blanca; === 75,08 pn 6

Pg.3+) 9 eptablesUnidadesAc AleatoriaN x = ( Distribuci0n binomiala ( )90,0100 B ( ( ) == 0901100 s Defectuosa 10b ( )90,0575 B ( ( ) == 90,01575 s Defectuosa 5,57 58

Pg.3!1 10 a ( )20,05 B ( ( ) ( ) ( ) ==== 50 8,02,0

051011 x P sana Penfrma P 672,0

b ( )20,010000 B ( ( )5200020,010000 >= pn ( ) ( )580001 >= pnqn 2000== np 40== npq ( Fe asume distribuci0n ( )402000 N

( ) ( ) ( ) ===


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70== np 75,665,035,0200 == npq Distribuci0n; ( )75,670 N ( ) ( ) ( ) ( ) =o para un particular >alor den 1+ q p q p = 21== q p De Moi>re e7ige; 5np 10n ( )432 N = 2132 n 64=n == 212164 4

Pg.3!) 14 a & . 1 5 / - y ! 0,. 0,' 0, 0,. 0,' b & i pi & i pi & i . pi . 0,. 0, 0,-

1 0,' 0,1 0, 5 0, . '0 / 0,. ',. ',. - 0,' 0,- /, 1 4,7 25,3 Media; = 7,4 iaci0n; == 21,3 79,1

Pg.3!) 15 a ( ) 125,0. == const x P y 4 9,,3,2 = xb & i * . 1 5 / 3 - pi * '+- '+- '+- '+- '+- '+- '+- '+-

& i pi * .+- 1+- +- 5+- /+- 3+- -+- +- 44/8 & i . pi * +- +- '/+- .5+- 1/+- +- / +- -'+- 284/8

Media; == 844 5,5 Des>iaci0n; == 25,55,54284 2 29,2)@'1


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c ( ) ( ) ( ) ( ) ( ) =+++==+=+=+== 8181818198766 x P x P x P x P x P 21 5,0= ( ) ( ) ( ) =+==+==iaci0n; === 21,13.25,6 2 1,1

Pg.3!) 18 cretaeatoriaDisVariableAl x = ( Distribuci0n ( )25,010 B

a ( ) ( ) === 644104 75,025,021025,0125,04

104 x P 146,0

b ( ) ( ) = == 100100 75,01125,0125,0

0

100 x P 056,0

*@'1


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c ( ) ( ) =

== 125,0125,0125,010

1010 10101010 x P 000000954,0

d ( ) ( ) ( ) ( ) ==+=+== 10988 x P x P x P x P

++= 0101928

75,025,010

1075,025,09

1075,025,08

10000416,0

e ( ) ( ) ( ) ( ) ( ) ==+=+=+== 32103 x P x P x P x P x P

6@'1


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( ) ( ) ++==== 5545355,05,05,053 55

6

3

6

3

ii

ii x P x P 5,0

Pg.3!) 20 Aciertos N x = ( Distribuci0n binomial; ( )25,010 B

( ) ( ) ( )

=== 8765 x P x P x P x P

+@'1


9/21


++= 087265,05,0885,05,0785,05,068145,0

G( ) ( ) +==+= 4453 5,05,04

85,05,0

3

843 x P x P 492,0

Pg.3!) 22 a aFavor N x = ( Distribuci0n binomial; ( )2,05 B ( ) ( ) ( ) ( ) ==+=+== 5433 x P x P x P x P

!@'1


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++= 0514238,02,0558,02,0458,02,0350579,0

b aFavor N x = ( Distribuci0n binomial; ( )2,020 B

( ) == 0115,0118,02,0

0

200 200 x P 0115,0

Pg.3!) 23 reversos N x = ( Distribuci0n binomial; ( )5,06 B

a ( ) ( ) = == 6363 5,0205,015,0

3

63 x P 3125,0

b ( ) ( ) ( ) ( ) ( ) ==+=+=+== 65433 x P x P x P x P x P

1,@'1


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+++= 65,066564636656,0

Pg.3!) 24 arcados Art culos! x = ( Distribuci0n binomial; ( )9,010 B

G ( ) ( ) ( )

=== 101010 0919,010101101lg x PdounoNo!arca A P 651,0

G

( ) ( )

== 4104

9,019,04

10

4 x P 000138,0

Pg.3!) 25 ( )75,04 B ( Media; == 75,04 pn 3=

11@'1


12/21


( ) ( ) ( )

+

==+== 0413

25,075,04

425,075,03

4433 x P x P x P

7383,0

Pg.3!) 26 [ ] " x" dx senx" == 2cos 00

12 " 21="

==+==

3

4

3

4

3 2cos0

21

214

3 xdxdx senxdx senx x P + 2

3121 933,0

Pg.3!* 27 a Bene&icio = 3660 24 "# / ( 240> 24240> "# "# 10>

( ) ==> 30

10

330

10

2

39000

1

900010

xdx

x x P 963,0

b Media@d?a; === 30

0

430

0

2

49000

1

9000

xdx

x x 5,22 d a"# = 245,22 540

Pg.3!* 28 No puede ser uncin de densidad . Para todo >alor de x es ( ) 0 x f

Pg.3!* 29 omo nada se especi&ica5 se toman inter>alos simCtricos respecto de la media. ( ) = " x" P ( ) ( ) 21 => " x P ( ) ( ) 21 += " x P

a ( ) =+= 75,02

5,01" x P 675,0=" ( ) 5,0675,0675,0 = x P en ( )10 N

b ( ) =+= 975,0

2

95,01" x P 96,1=" ( ) 95,096,196,1 = x P en ( )10 N

c ( ) =+

= 995,0299,01

" x P 575,2=" ( ) 99,0575,2575,2 = x P en ( )10 N

Pg.3!* 30 ( )cma Altura!edi x = ( Distribuci0n normal; ( )12176 N

( ) ( ) =


13/21


( ) ( ) ( ) ( ) ===

= 22,20022,29100100

910080

10080 z P z P z P z P x P

( ) ( )[ ] [ ] === 9868,015,022,210 z P z P 4868,0 ( %68,48

Pg.3!* 33 Datos Porcenta%e x = ( Distribuci0n ( )220 N

( ) ( ) ( ) ===>=

>=> 9332,015,115,12

202323 z P z P z P x P 0668,0

( ) ( ) ( ) ===> 5438,0111,0111,0

75,1

8,566 z P z P z P x P 4562,0 %6,45

b Admitidos N x = ( Distribuci0n binomial; ( )4562,010 B

( ) ( ) == 644104 5438,04562,02104562,014562,04

104 x P 235,0

Pg.3!* 39 Anversos N x = ( Distribuci0n binomial; ( )5,0900 B13@'1


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!uy*randen = 4505,0900 == pn ( )5> ( ) 4501 == pnqn ( )5>on De Moi>re se apro7ima mediante ( )15450 == npqnp N

( ) ( ) ( ) ( ) == ( ) 80311120 ==qn ( )5> ( umple De Moi>re Fe puede apro7imar por la distribuci0n "ormal; ( ) N

=== 31120 pn 40 ( ) === 38031131120q pn 16,5

( ) ( ) =

==

16,5405,45

16,5405,34

5,455,344535 z P ' P x P ( ) = 07,107,1 z P

( ) ( ) ( ) == 107,127,1007,1 z P z P z P 18577,02 7154,0

Pg.3!6 43 a

x personas que an ledo la novela * varia"le aleatoria discreta

6n particular, el n2 den 4 ami7os la a ledo con pro"a"ilidad ! 0,8 8a >ariable x pertenece a una distribucin binomial ; ( )8,04 B

1)@'1


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b G ( ) ( ) = == 242 8,018,0

2

42 x P 04,064,06 1536,0

G ( ) ( ) ( ) ( ) ==+=+== 4322 x P x P x P x P

=++= 0413222,08,0442,08,03

42,08,02

4

++ 4096,04096,01536,0 9728.0

Pg.3!6 44 8 9 :2 canastas * varia"le aleatoria discreta en distri"ucin; ( )26 N

a ( ) ( ) ( ) ===

>=> 6915,015,015,0

2

677 z P z P z P ' P 3085,0

b ( ) ( ) ( ) ( ) ===


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Pg.3!6 46 = x permetro torcico * varia"le aleatoria en distri"ucin ( )490 N a ( ) ( ) ( ) ==


17/21


Pg.3!6 49 = x >ateras defectuosas * 28=nq ( )5> 8a distribuci0n" se puede apro7imar por la ( )9,212 === npq N

( ) ( ) ( ) ( ) ==


18/21


G n la nue>a hip0tesis5 la distribuci0n es la binomial( )2,010000 B 20002,010000 == pn ( )5> ( ) 80002,0110000 ==qn ( )5> Fe cumple De Moi>re 9 se puede apro7imar con la distribuci0n normal; ( )( )402,0120002000 === npqnp N

( ) ( ) ( ) ==

==> 01,2

40

20005,20805,20802080 z P z P ' P x P

( ) endidos J =

150150600

3

Falario semanal K 1*, ( oches >endidos K =150

150150 0

n la semana >ende; Ms de 0 9 menos de 3 entre 1 ! 3 FegAn el teorema Ba9es5 se puede escribir;

( ) ( )( )( )

( )( )( )

==

=

=> 9192,014,114,1

5

105112112 z P z P z P ' P 0808,0

( ) ( ) ( ) =

>=> 9772,01212

5

105115115 z P z P z P P 0228,0

( ) ( )

( ) >>

=>> 0808,00228,0

112

115112115 P

P

' ' P 2822,0

Pg.3! 55 a = ' !eso yo7ures * z P

/ablas; ( ) 9798,005,2 = z P

( )0002,0=

( ) 9803,006,2 = z P ( )003,0=

Fe toma; 9798,0 05,2 ( 05,22

120 = 1,124= ( ) #

Pg.3! 56 l >alor de # se calcula mediante la e7presi0n; ( )[ ] 1213

1

2 = dx x"

( )[ ]3

434

321

3

1

33

1

2 " x x

" dx x" =+= 134 ="

43="

( ) ( )[ ] =+==> 3

5,1

3

5,1

32 34

343

2143

5,1 x x

dx x ' P 32

27 84375,0=

( ) =>>> 5,15,15,1 321 ' ' ' P ( ) ( ) ( ) ==>>> 3321 84375,05,15,15,1 x P ' P ' P 6007,0

Pg.3! 57 a ( ) =

=== 4

11

4

333,0

43,033,0

43

43

3

1

43

3

1

2 x xdx x x x

== 63,0 8,1 )( 1oras

b ( ) ( ) =

==>< 4

5,15,1

4

223,033,05,12

43

43

2

5,1

2 dx x x x ' ' P

= = 890625,13,04

5,2375,343,0 5672,0

Pg.3! 58 edia; = 3


20/21


21/21


== 45 8,02,058,01 = 4096,03277,01 2627,0

Unidad 16 - Distribuciones de Probabilidad, problemas resueltos

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