distribuciÓn de probabilidades discretas y continua

7/21/2019 DISTRIBUCIÓN DE PROBABILIDADES DISCRETAS Y CONTINUA

http://slidepdf.com/reader/full/distribucion-de-probabilidades-discretas-y-continua 1/25

DOCENTE: ING. MIGUEL MALDONADO

SEMESTRE:

SEGUNDO “A”

RESPONSABLE:

PESÁNTEZ CAPABRUNO DAVID

AÑO LECTIVO:

2015 – 2016

UNIVERSIDAD TÉCNICADE MACHALA

UNIDAD ACADÉMICA DEINGENIERIA CIVIL

EST

ADÍSTI

CA



INTRODUCCIÓN

U! "#$%&#'()#* "+ ,&-'!'##"!" #"#)! %-"! ! /!! "+ !-&+$ (+ ,(+"+&+,&+$+%!&$+ )-- &+$(%!"- "+ ( +3,+&#+%-. U! "#$%&#'()#* "+ ,&-'!'##"!" +$$##!& ! "#$%&#'()#* "+ 4&+)(+)#!$ &+!%#!$ .S# +'!&/- + + "+ "+$)&#'#& +,!$!"- "+$)&#'+ ! ,&-'!'##"!" (+ ( ++%- $+ &+!#)+ + + 4(%(&- )-$%#%(7+ (!8+&&!#+%! 4("!+%! ,!&! ! ,&-$,+)%#! ,(+$%- (+ $+ ,(+"+ "#$+9!& (+$)+!&#- "+ !)-%+)##+%-$ 4(%(&-$ )-$#"+&!"- !$ %+"+)#!$ !)%(!+$ "+ "#+&$-$4+*+-$ !%(&!+$.

L!$ "+)#$#-+$ +$%!"$%#)!$ '!$!"!$ + ! +$%!"$%#)! #4+&+)#! $- 4("!+%!+$+ ! #+$%#/!)#* (+ $- +!(!"!$ + %;&#-$ "+ "#$%&#'()#* "+ ,&-'!'##"!"+$.

E + ,&+$+%+ %&!'!<- $+ +$%("#! "+ !+&! =/# -$ "#+&$- %#,-$ "+ "#$%&#'()#*,&-'!'#$%#)! )!&!)%+&#!&+-$ )!"! "#$%&#'()#* ! 4("!+%!)#* !%+=%#)! "+-$ "#+&$-$ &+$(%!"-$ - $+ +4-)!&! + + ,&+$+%+ %&!'!<-> $*- + ##%!&; !+$%("#- "+$)&#,%#- "+ ! "#$%&#'()#* "+ ,&-'!'##"!"+$ "#$)&+%!$.



OBJETIVOS

• I"+%#4#)!& -$ )-)+,%-$ "+ 4()#* "+ ,&-'!'##"!" 7 4()#* "+ "+$#"!"

,!&! !&#!'+$ !+!%-&#!$ "#$)&+%!$ 7 )-%#(!$ &+$,+)%#!+%+.

• R+)--)+& ! "#$%&#'()#* '#-#! 7 + )-%+3%- "+ $( !,#)!)#*.

• C!)(!& ,&-'!'##"!"+$ ,!&! !&#!'+$ !+!%-&#!$ '#-#!+$.

• R+)--)+& ! "#$%&#'()#* -&! 7 + )-%+3%- "+ $( !,#)!)#*.

• C!)(!& ,&-'!'##"!"+$ ,!&! !&#!'+$ !+!%-&#!$ -&!+$ +"#!%+

%#,#4#)!)#* 7 ($- "+ %!'!$.

• I"+%#4#)!& )(="- +$ !)+,%!'+ !,&-3#!& (! "#$%&#'()#* '#-#! ,-& (!

-&! 7 )*- 8!)+&-.



Tabla de contenidoDISTRIBUCIONES DE PROBABILIDAD DISCRETAS........................................................5

DISTRIBUCIÓN DISCRETA UNIFORME.......................................................................5

DISTRIBUCION BINOMIAL.........................................................................................DISTRIBUCIÓN BINOMIAL NE!ATI"A........................................................................#

DITRIBUCION !EOMETRICA................................................................................... $%

DISTRIBUCION &IPER!EOMETRICA........................................................................$$

DISTRIBUCION POISSON........................................................................................ $'

DISTRIBUCI(N CONTINUA.........................................................................................$)

DISTRIBUCION CONTINUA NORMAL.......................................................................$*

DISTRIBUCIÓN NORMAL ESTANDAR.......................................................................$*

E+tanda,i-aci(n de la Di+t,ibci(n No,/al............................................................$5FISTRIBUCION !AMMA...........................................................................................$0

DISTRIBUCION 1I2CUADRADO.................................................................................'%

DISTRIBUCIÓN BETA.............................................................................................. '$

DISTRIBUCIÓN DE 3EIBULL...................................................................................''

Biblio4,a6a............................................................................................................... '*

Tabla de il+t,acione+

Il+t,aci(n $2 Di+t,ibci(n Unio,/e...........................................................................Il+t,aci(n '2 Di+t,ibci(n Bino/ial............................................................................7Il+t,aci(n )2 Di+t,ibci(n Contina Unio,/e..........................................................$)Il+t,aci(n *2 Di+t,ibci(n No,/al 8a,a 9a,io+ 9alo,e+ de : ; <..............................$*Il+t,aci(n 52 Di+t,ibci(n No,/al E+t=nda,.............................................................$5Il+t,aci(n 2 Tabla de E+tanda,i-aci(n de la Di+t,ibci(n No,/al............................$5Il+t,aci(n 02 Di+t,ibci(n No,/al ; la Di+t,ibci(n No,/al E+t=nda,......................$

Il+t,aci(n 7 2 Con9e,+i(n a Di+t,ibci(n No,/al......................................................$0Il+t,aci(n #2 !,=>co de la nci(n de den+idad 8a,a el e?e/8lo.............................$7Il+t,aci(n $%2 P,obabilidad de e?e/8lo e+ i4al @,ea 8intada.................................'%Il+t,aci(n $$2 !,=>co de la di+t,ibci(n 1i2Cad,ado con 95.................................'$Il+t,aci(n $'2 @,ea de 8,obabilidad del lite,al a......................................................''

DISTRIBUCIONES DE PROBABILIDAD DISCRETAS

S- M-"+-$ !%+=%#)-$ ,!&! )!)(!& ! ,&-'!'##"!" + !/(-$ ,&-'+!$ %,#)-$+ -$ (+ #%+&#++ !&#!'+$ !+!%-&#!$ "#$)&+%!$.



E -'<+%#- +$ -'%++& (! 4*&(! !%+=%#)! f ( x) ,!&! "+%+&#!& -$ !-&+$ "+

,&-'!'##"!" "+ ! !&#!'+ !+!%-&#! X

Modelos de distribucioes de !rob"bilid"d de #"ri"bles discret"s• Unio,/e• Bino/ial• Bino/ial Ne4ati9a• !eo/t,ica• &i8e,4eo/t,ica• Poi++on

DISTRIBUCIÓN DISCRETA UNI$ORMEDe%iici&'

S+! X: !&#!'+ !+!%-&#! "#$)&+%!> (? (1 (2 (@ ( S- -$ !-&+$

(+ ,(+"+ %-!& X )- #/(! ,&-'!'##"!"E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" X +$

f ( x) ) {1

n x= x1, x2,…,x n

0,otro x

So!orte

S+ -'%#++ "#&+)%!+%+ "+ !$ "+4##)#-+$ )-&&+$,-"#+%+$

S+! : V!&#!'+ !+!%-&#! )- "#$%&#'()#* "#$)&+%! (#4-&+

Medi" * V"ri"+" De L" Distribuci& Discret" Ui%or,e

Medi" µ= E ( X )= Σ X xf ( X )= Σi=1

n xi f ( xi )=

1

n Σ i=1

n xi

V"ri"+"

i−¿ μ

x¿¿¿

σ 2= E [ ( X − μ )2]= Σ X ( xi− μ)2 f ( x)=1

n Σ i=1

n ¿

E-e,!lo'

U +3,+&#+%- )-$#$%+ + !!& ( "!"- + &+$(%!"-. S# +$ ! !&#!'+ !+!%-&#!)-&&+$,-"#+%+ ! -$ $+#$ &+$(%!"-$ ,-$#'+$ +)(+%&+ $( "#$%&#'()#* "+,&-'!'##"!".

R+$,(+$%!



C!"! &+$(%!"- %#++ #/(! ,&-'!'##"!" ,-& - %!%- ! "#$%&#'()#* "+,&-'!'##"!" "+ +$ "#$)&+%! (#4-&+.

P( X = x)=f ( x )={1

6, x=1,2,… ,6

0, para todo x

S-,-&%+

1 2 @ 56

C!)(+ ! ,&-'!'##"!" (+ %-+ + !-& @

PF?@ ?4 F@ ?1H6

x i

μ= Σ x i f ¿ ?1F1H6 2F1H6 @F1H6 F1H6 5F1H6 6F1H6 [email protected]

J2?K12F1H622F1H6@[email protected]

J2?1H612.25?2.16

G&=4#)- "+ "#$%&#'()#* "#$)&+%! (#4-&+

Ilustración 1- Distribución Uniforme

DISTRIBUCION BINOMIALDe%iici&'

S+! X: V!&#!'+ !+!%-&#! "#$)&+%! F&+,&+$+%! ! )!%#"!" "+ +$!7-$)-$#"+&!"-$ “;3#%-$” + (! $+&#+ "+ +$!7-$ &+!#!"-$.

?0 12 !-&+$ (+ ,(+"+ %-!&

P: P&-'!'##"!" "+ (+ + &+$(%!"- "+ )!"! +$!7- $+! “;3#%-”

E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ +$



f ( x )=(n

x) p x (1− p )

n− x, x=0,1,2, … , n

De,ostr"ci&

A &+!#!& +$!7-$ $+ -'%#++ ( ;3#%-$ 7 . ( 4&!)!$-$ ,-& - %!%- $#+"- +$!7-$

#"+,+"#+%+$ ! ,&-'!'##"!" "+ -'%++& +$%-$ &+$(%!"-$ +$ p x (1− p)n− x

.P+&- +

-$ +$!7-$ &+!#!"-$ 8!7 (n

x) 4-&!$ "#4+&+%+$ "+ -'%++& -$ ( ;3#%-$ 7 -$ /(

4&!)!$-$. E$%+ +&- +$ +%-)+$ ( 4!)%-& ,!&! + !-& "+ ,&-'!'##"!" !%+&#-&.

L-$ $'--$ (n

x) C xn C x

n

&+,&+$+%! + +&- "+ )-'#!)#-+$ -

!&&+/-$ "#4+&+%+$ (+ $+ -'%#++ )- +++%-$ "+ -$ )(!+$ $+ %-! (+++%-$.

E-e,!lo'

U! 4='&#)! %#++ (! -&! "+ )!#"!" )-$#$%+ + ++/#& ! !!& "#!&#!+%+20!&%#)(-$ ,&-"()#"-$ 7 "+%+&#!& + +&- "+ (#"!"+$ "+4+)%(-$!$ $# 8!7 "-$ -=$ !&%)(-$ "+4+)%(-$-$ ! 4!'&#)!)#* $+ "+%#++ ,!&! #$,+))#* "+ -$ +(#,-$. S+

)--)+ ,-& +3,+&#+)#! (+ ! ,&-'!'##"!" "+ (+ ( !&%)(- ,&-"()#"- $+!"+4+)%(-$- +$ 5. E)(+%&+ ! ,&-'!'##"!" "+ (+ + )(!(#+& "! ! ,&-"())#* $+"+%+/! ! !,#)!& +$%! -&! "+ )-%&- "+ )!#"!".

R+$,(+$%!

E$%! $#%(!)#* )-&&+$,-"+ ! ( +3,+&#+%- '#-#!

N?20 )!%#"!" "+ +$!7-$ FI"+,+"#+%+$

P?0.05 P&-'!'##"!" "+ ;3#%- FC-$%!%+: V!&#!'+ !+!%-&#! "#$)&+%! F)!%#"!" "+ !&%)(-$ "+4+)%(-$-$

? 0 1 2. 20 !&#!'+ !+!%-&#! "#$)&+%! F)!%#"!" "+ !&%)(-$"+4+)%(-$-$

P ( X = x )= f ( x )=(n

x ) p x (1− p)n− x=(20 x )0.05 x (0.95)20− x

, x=0,1,2,3,… ,20

E%-)+$

P ( x ≥2 )=1− P ( X ≤1 )



¿1−( P ( x=0 )+ P ( x=1 ) )=1−(f (0 )+ f (1))

f (0)=(200 )0.050(0.95)20=0.3585

f (1)=(201 )

0.0510.9519=0.3774

P ( X ≥2 )=1−0.3585−0.3774=0.2641=26.41

Medi" * V"ri"+" De L" Distribuci& Bio,i"l

S+! X' !&#!'+ !+!%-&#! "#$)&+%! )- "#$%&#'()#* '#-#! )- ,!&=+%&-$ 0 !

E%-)+$

Medi" µ= E( x )=np

V"ri"+"σ

2=V ( x)=np(1− p)

E-e,!lo'

E)(+%&+ ! +"#! 7 ! !&#!! ,!&! + +<+,- "+ )-%&- "+ )!#"!" + ! 4='&#)!.

R+$,(+$%!:

µ=np=20(0.05)=1

σ 2

=npq=20(0.05)(0.95)=0.95

1 &+,&+$+%! ! )!%#"!" "+ !&%)(-$ "+4+)%(-$-$ (+ $+ -'%#++ )!"! "!

σ 2

+$ (! +"#! "+ ! !&#+"!" - "#$,+&$#* "+ -$ !-&+$ "+ .

G&=4#)- "+ ! "#$%&#'()#* '#-#!

L! "#$%&#'()#* '#-#! $( /&=4#)- )- (! 4-&! $#;%&#)! )(!"- ,?0.5

E-e,!lo'G&!4#(+ ! "#$%&#'()#* '#-#! )- ?10 ,?0.5



f ( x)=(10 x )0.5 x0.5

10− x=(10 x ) (0.5)10 , x=0,1,2,… ,10

f (0)=0.0010 f (8)=0.0439

f (1

)=0.0098

f (9

)=0.0098

f (2)=0.0439 f (10)=0.0010

Ilustración 2- Distribución Binomial

DISTRIBUCIÓN BINOMIAL NE2ATIVADe%iici&'

S+! : V!&#!'+ !+!%-&#! "#$)&+%! )- "#$%&#'()#* '#-#! +/!%#! F)!%#"!" "++$!7-$ &+!#!"-$ 8!$%! -'%++& Q “3(itos”P: P&-'!'##"!" "+ “;3#%-”. +$ ( !-& )-$%!%+ + )!"! +$!7-? QQ1Q2F!-&+$ (+ ,(+"+ %-!& ! !&#!'+ 3E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ X +$'

P ( X = x )=f ( x )=

(

x−1

k −1

) p

k (1− p) x−k , x=k , k +1,k +2,…

Medi" * V"ri"+" de l" Distribuci& Bio,i"l Ne4"ti#"

Medi": ¿ E [ X ] k

p

V"ri"+": σ 2=V [ X ]= K

P ( 1

P −1)

E-e,!lo'



S(,-#+"- (+ ! ,&-'!'##"!" "+ (+ (! ,+&$-! )-%!/#! )#+&%! +4+&+"!" ! !(+ +$%= +3,(+$%! +$ @0 )!)(+ ! ,&-'!'##"!" (+ ! ";)#! ,+&$-! +3,(+$%! ! !+4+&+"!" $+! )(!&%! + )-%&!+&!.

R+$,(+$%!

C!"! ,+&$-! +3,(+$%! ! ! +4+&+"!" )-$%#%(7+ ( +$!7- +$%-$ +$!7-$$- #"+,+"#+%+$ 7 ! ,&-'!'##"!" "+ “;3#%-” +$ )-$%!%+: [email protected]%+ (+“;3#%-” - $#+,&+ %#++ (! )--%!)#* 4!-&!'+

P-& ! ,&+/(%! )-)(#-$ (+ ! !&#!'+ "+ #%+&;$ X %#++ "#$%&#'()#*'#-#! +/!%#! )- Q? ,?0.@

S+! X' )!%#"!" "+ +$!7-$ &+!#!"-$ 8!$%! -'%++& Q “;3#%-$” F!&#!'+!+!%-&#! "#$)&+%!

? 56

P ( X = x )= f ( x )=( x−1

4−1)0.3

4

(1−0.3) x−4

, x=4,5,6,…

P-& - %!%-

P ( x=10)=f (10 )=(10−1

4−1 )0.340.7

10−4=0.08

DITRIBUCION 2EOMETRICADe%iici&'

S+! : !&#!'+ !+!%-&#! "#$)&+%! )- "#$%&#'()#* /+-;%&#)! F)!%#"!" "++$!7-$ &+!#!"-$ 8!$%! -'%++& + ,&#+& “;3#%-”

?1 2@ FV!-&+$ 4!)%#'+$ ,!&! ! !&#!'+ XP: ,&-'!'##"!" )-$%!%+ "+ “;3#%-” + )!"! +$!7-

E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ X +$:

P( X = x)=f ( x )= p(1− p) x−1

3?1 2@

L" Medi" * V"ri"+" de l" Distribuci& 2eo,3tric"

Medi"' μ= E [ X ]= 1 P

,



V"ri"+"' σ 2=V [ X ]= 1

p(1

p−1)

E-e,!lo'

C!)(+ ! ,&-'!'##"!" (+ + + (#%- !!#+%- "+ %&+$ -+"!$ $+ -'%+/! %&+$$+-$ ,-& ,&#+&! +.

R+$,(+$%!

E + +3,+&#+%- "+ !!& %&+$ -+"!$ 8!7 &+$(%!"-$ ,-$#'+$.

E )!"! +$!7- ! ,&-'!'##"!" (+ $!/! %&+$ $+-$ +$ )-$%!%+ + #/(! ! 1H7 ! ,&-'!'##"!" (+ - $!/! %&+$ $+-$ +$ H.

E$%-$ +$!7-$ $- #"+,+"#+%+$ 7 ,-& ! ,&+/(%! )-)(#-$ (+ ! !&#!'+"+ #%+&;$ X %#++ "#$%&#'()#* /+-;%&#)! )- !)5678

S+! X' )!%#"!" "+ +$!7-$ 8!$%! -'%++& + ,&#+& “;3#%-” F!&#!'+ !+!%-&#!"#$)&+%!

?1 2@

P ( X = x )= f ( x )=( 18 )( 78 ) x−1

, x=1,2,3,…

P-& - %!%-

P ( X =5 )=f (5)=(18 )(

78 )

5−1

=0.0733

DISTRIBUCION 9IPER2EOMETRICA

De%iici&'

S+!

N' C!%#"!" "+ +++%-$ "+ )-<(%- "+ (+ $+ %-! ! (+$%&!

:: C!%#"!" "+ +++%-$ +3#$%+%+$ (+ $+ )-$#"+&! “;3#%-$”

N' T!!9- "+ ! (+$%&!

X: V!&#!'+ !+!%-&#! "#$)&+%! F+$ ! )!%#"!" "+ &+$(%!"-$ )-$#"+&!"-$ “;3#%-$”(+ $+ -'%#++ + ! (+$%&!

(?0 12 F$- -$ !-&+$ (+ ,(+"+ %-!& X

E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ X +$



f ( x )=(k

x )( N −k

n− x )( N

n ) , x=0,1,2,… ,n

Medi" * V"ri"+" de l" Distribuci& 9i!er4eo,3tric"

Medi": μ= E [ X ]=n K

N ,

V"ri"+": σ 2=V [ X ]=nK

N (1− K

N )( N −n

N −1 )

E-e,!lo'

U! )!<! )-%#++ '!%+&!$ "+ !$ )(!+$ +$%= + '(+ +$%!"- 7 !$ &+$%!%+$"+4+)%(-$!$. S+ %-! (! (+$%&! +#/#+"- ! !!& %&+$ '!%+&!$. C!)(+ !,&-'!'##"!" (+ + ! (+$%&! $+ -'%+/!.

! N#/(! '!%+&! + '(+ +$%!"-' A +-$ (! '!%+&! + '(+ +$%!"-) C!)(!& ! +"#! 7 !&#!!

R+$,(+$%!E$%+ +$ ( +3,+&#+%- "+ (+$%&+- $# &++,!- ,-& - %!%- +$ ( +3,+&#+%-

8#,+&/+-;%&#)- )-N); <T-%! "+ +++%-$ "+ )-<(%-=:)> <T-%! "+ +++%-$ )-$#"+&!"-$ “;3#%-$”=)? <%!!9- "+ ! (+$%&!=X' )!%#"!" "+ '!)%+&#!$ + '(+ +$%!"- + ! (+$%&! F!&#!'+ !+!%-&#!

E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" +$

f ( x )=( K

x )( N − K

n− x )( N

n ) =

(4 x)(9−4

3− x )(93)

, x=0,1,2,3

! P ( x=0 )= f (0 )=(40)(

9−4

3−0)(93)

=0.119

' P ( x ≥1 )=1− P ( x<1 )=1− f (0 )=1−0.119=0.881

) μ=3 (4

9 )=1.333



σ 2=

3 (4 )9 (1−4

9 )( 9−3

9−1 )=0.555

DISTRIBUCION POISSONDe%iici&'

S+! X: V!&#!'+ !+!%-&#! "#$)&+%! )- "#$%&#'()#* "+ P-#$$- F)!%#"!" "+“;3#%-$” + (! &+/#* - #%+&!- +$,+)#4#)!"-$? 012F!-&+$ ,-$#'+$ ,!&! ! !&#!'+ X

λ : )!%#"!" ,&-+"#- "+ “;3#%-$” + ! &+/#* - #%+&!- +$,+)4#)-$

E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ X +$:

f ( x )=e− λ

λ x

x ! , x=0,1,2,… , e=2.718281828

Medi" * V"ri"+" De L" Distribuci& De Poisso

Medi": μ= E [ X ]= λ

V"ri"+": VK?

E-e,!lo'

L! )!%#"!" "+ +&&-&+$ "+ %&!$#$#* "+ "!%-$ + (! 8-&! +$ 5 + ,&-+"#-.S(,-#+"- (+ +$ (! !&#!'+ )- "#$%&#'()#* "+ P-#$$- "+%+&#+ ! ,&-'!'##"!"

(+:! E )(!(#+& 8-&! -)(&&! $-!+%+ 1 +&&-& ' E )(!(#+& 8-&! -)(&&! ! +-$ @ +&&-&+$

R+$,(+$%!S+! X' V!&#!'+ !+!%-&#! "#$)&+%!F)!%#"!" "+ +&&-&+$ ,-& 8-&! ?5 F,&-+"#-"+ +&&-&+$ "+ %&!$#$#* + 1 8-&!

"= P ( x=1 )=f (1 )= e−55

1

1! =0.0337

b= P ( x ≥3 )=1− P ( x ≤2 )=1−(f (0 )+ f (1 )+ f (2 ) )=1−0.1247=0.8743

DISTRIBUCIÓN CONTINUA



S- !(+!$ + !$ (+ ! 4()#* "+ "#$%&#'()#* +$ (! 4()#* )-%#(!. S+)-&&+$,-"+ )- + ,&#+& %#,- "+ /&=4#)! #$%-.

G++&!+%+ $+ )-&&+$,-"+ )- !&#!'+$ !$-)#!"!$ ! +3,+&#+%-$ + -$ )(!+$! !&#!'+ +"#"! ,(+"+ %-!& )(!(#+& !-& + ( #%+&!-> +"#)#-+$ '#-;%&#)!$,-& +<+,-.

U )!$- ,!&%#)(!& "+%&- "+ !$ !&#!'+$ !+!%-&#!$ )-%#!$ 7 ! )(! ,+&%++)+%-"-$ -$ +<+,-$ ($(!+%+ (%##!"-$ $- !$ "+-#!"!$ !&#!'+$ !+!%-&#!$!'$-(%!+%+ )-%#(!$.

DISTRIBUCIÓN UNI$ORME CONTINUADe%iici&'

S+! : V!&#!'+ !+!%-&#! )-%#(! %#++ "#$%&#'()#* +$%= "!"! ,-&

f ( x )=

{

1

b−a, a ≤ x ≤ b

0 , paraotro x

A' $- -$ ,!&=+%&-$ ,!&! +$%+ -"+-

R+,&+$+%!)#* /&=4#)! "+ ! "#$%&#'()#* (#4-&+ )-%#(!

Ilustración 3- Distribución Continua Uniforme

Medi" * V"ri"+" de l" Distribuci& Ui%or,e Cotiu"Medi": μ= E ( x )= 1

2(a+b)

V"ri"+": σ 2=V ( x )=

1

12(b−a)2

E-e,!lo'C(!"- 4!! )#+&%! )-,-+%+ "+ (! =(#! +$%! "+'+ "+%++&$+ 8!$%! (+ $+!&+,!&!"-. S(,-#+"- (+ + %#+,- "+ &+,!&!)#* ,(+"+ %-!& )(!(#+& !-& +%&+ 17 5.

"= C"lcule l" !rob"bilid"d @ue l" dur"ci& to,e "l ,eos or"s8

S-()#*



: !&#!'+ !+!%-&#! )-%#(! F"(&!)#* "+ ! &+,!&!)#* %#++ "#$%&#'()#*(#4-&+ ,-& - %!%- $( 4()#* "+ "+$#"!" +$

f ( x )= 1

b−a=

1

5−1=

1

4,1≤ x ≤5

P ( x ≥2 )=∫2

5

14

dx=34=75

b= C"lcule el #"lor es!er"do de l" dur"ci& de l" re"!"rici&

E ( x )=1

2(a+b )=

1

2(1+5 )=3ora!

DISTRIBUCION CONTINUA NORMALDe%iici&:

S+! X' V!&#!'+ !+!%-&#! )-%#(! )- +"#! 7 !&#!! σ 2 X %#++

"#$%&#'()#* -&! $# $( 4()#* "+ "+$#"!" +$:

f ( x )= 1

σ √ 2" e

−1

2( x− μ

σ )2

,−#< x<+#

L! /&!4#)! "+ 4 +$ $##!& ! ,+&4# "+ )-&%+ +&%#)! "+ (! )!,!! 7 %#++ !$$#/(#+%+$ )!&!)%+&$%#)!$:

1 E$ $#;%&#)! !&+"+"-& "+ 1 2 S( !$%-%! +$ + +<+ 8-&#-%!

@ S($ ,(%-$ "+ #4+3#* +$%= ('#)!"-$ + J 7 J

Ilustración 4- Distribución Normal para varios valores de μ !

DISTRIBUCIÓN NORMAL ESTANDARDe%iici&'

S+! ' V!&#!'+ !+!%-&#! )-%#(! )- +"#! ?0 7 !&#!! J2?1 %#++"#$%&#'()#* N-&! E$%="!& $# $( 4()#* "+ "+$#"!":

f ( x )= 1

√ 2" e−1

2 $

2

,−#< $<+#



f ( t ) dt =¿∫−#

$1

√ 2" e

−1

2 t

2

dt ,−#< $<+#

% ( x )= P ( & ≤ $ )=∫−#

$

¿

Ilustración "- Distribución Normal #st$ndar

E-e,!lo'U$!"- ! %!'! "+ ! "#$%&#'()#* -&! +$%="!& )!)(+:

! PFZ 1.5PFZ 1.5? 4F1.5? 0.265

E &+$(%!"- $+ %-&! "#&+)%!+%+ "+ ! %!'! "+ "#$%&#'()#* -&! +$%="!&:

Ilustración %- &abla de #standari'ación de la Distribución Normal

Est"d"ri+"ci& de l" Distribuci& Nor,"lNot"ci&

W NFJ D+4#+ ! )-- (! !&#!'+ )- "#$%&#'()#* )- +"#! 7"+$#!)#* +$%="!& J



ZWNF01 D+4#+ ! Z )-- (! !&#!'+ )- "#$%&#'()#* -&! +$%="!& )-+"#! 0 7 "+$#!)#* +$%="!& 1

De%iici&' E$%!"!&#!)#* "+ ! D#$%&#'()#* N-&!S+! (! !+!%-&#! )- "#$%&#'()#* -&!: WN F J. E%-)+$ ! !&#!'+

!+!%-&#!& =

X −µ

σ T#++ "#$%&#'()#* -&! +$%="!&: ZWNF01

Re!reset"ci& 4r"%ic"

Ilustración (- Distribución Normal la Distribución Normal #st$ndar

L! &+!)#* +%&+ 3 7 +$ #+! ,-& - %!%- ! "#$%&#'()#* "+ "+'+ %++& (! 4-&!$##!& ! ! "#$%&#'()#* N-&! M+"#!%+ !$ "+4##)#-+$ "+ !-& +$,+&!"- 7 !&#!!:

E (& )= E ( x− μ

σ )= 1σ [

E ( x )− E ( μ ) ]= 1σ

( μ− μ )=0

V ( & )=V ( x− μ

σ )= 1

σ 2 (σ

2−0 )=1

S+ ,(+"+ ,&-'!& (+ Z %#++ "#$%&#'()#* -&! +$%="!& ZWNF01

E<+,-S+! WN F10 J. E)(+%&+ J %! (+ PF ? 0.025S-()#*



Ilustración ) - Conversión a Distribución Normal

P ( x ≤9 )= P (& ≤ $ )= % ( $)=0.025⟹ $=−1.96

& = x− μ

σ ⇒−1.96=

9−10

σ ⇒ σ =0.5102

$ISTRIBUCION 2AMMAS+! : V!&#!'+ !+!%-&#! )-%#(! X %#++ "#$%&#'()#* GAMMA $# $( 4()#* "+"+$#"!" +$

f ( x )={ 1

'(

) (( ) x

( −1e

− x

' , x>0

0 , para otro x

( >0 , '>0 on*o para+etro para ete +ode*o

Medi" * V"ri"+" !"r" l" Distribuci& 2",,"S+! X (! !&#!'+ !+!%-&#! )-%#(! )- "#$%&#'()#* G!! +%-)+$

Medi"' μ= E ( x )=('

V"ri"+"' σ 2=V ( x )=( '

2

E-e,!lo'



E %#+,- + 8-&!$ (+ $+!!+%+ &+(#+&+ (! =(#! ,!&! !%+##+%- +$

(! !&#!'+ !+!%-&#! )- "#$%&#'()#* /!! )- ,!&=+%&-$ ( =3, '=2

! E)(+%&+ ! ,&-'!'##"!" (+ !/(! $+!! + %#+,- "+ !%+##+%- $+!!7-& ! 8-&!$.

S-()#*:S+! X: "(&!)#* "+ !%+##+%- + 8-&!$ F!&#!'+ !+!%-&#! $( "+$#"!" "+,&-'!'##"!" +$:

f ( x )= 1

'( ) (( )

x( −1

e− x

' = 1

23 ) (3 )

x3−1

e− x

2 = 1

16 x

2e− x

2

Ilustración *- +r$,co de la función de densidad para el eemplo

! PFX +$ + =&+! &+$!%!"! + + /&!4#)-

P ( X >8)=1 P ( X ≤ )=1 1

16

∫0

8

x2

e

− x

2 dx

P!&! #/&+$!& $+ ,(+"+ !,#)!& "-$ +)+$ ! %;)#)! "+ #%+/&!)#* ,-& ,!&%+$:

∫ x2

e

− x

2

"3

-= x 2⇒d-=2 x dx

d.=e− x

2 dx ⇒.=−2e− x

2

¿−2 x2 e

− x

2 +4∫ x e

− x

2 dx



∫ x e− x

2 dx

-= x ⇒d-=dx

d.=e− x

2dx

⇒ .=−2e− x

2

¿−2 x e− x2 +2∫e

− x2 dx

S($%#%(7+"- -$ &+$(%!"-$ #%+&+"#-$

P ( x>8 )=1− 1

16[−2 x

2e

− x

2 +4 (−2 xe

− x

2 +2(−2e

− x

2 ))]80=0.2381

DISTRIBUCION EPONENCIAL

D+4##)#*S+! X' V!&#!'+ !+!%-&#! )-%#(! X %#++ "#$%&#'()#* E3,-+)#! $# $(

"+$#"!" "+ ,&-'!'##"!" +$%! "!" ,-& f ( x )={ 1

' e

− x

2 x>0

0 , paraotro x

E "-"+ Y X 0 +$ + ,!&=+%&- ,!&! +$%+ -"+-

Medi" * #"ri"+" !"r" l" distribuci& e(!oeci"l

S+! X' V!&#!'+ !+!%-&#! )-%#(! )- "#$%&#'()#* +3,-+)#!Medi"' μ= E ( x )= '

V"ri"+!: σ 2=V ( x )= '

2

E-e,!lo'U $#$%+! ($! ( )-,-+%+ )(7! "(&!)#* + !9-$ +$ (! !&#!'+ !+!%-&#! )-"#$%&#'()#* +3,-+)#! )- +"#! "+ !9-$. S# $+ #$%!! @ "+ +$%-$ )-,-+%+$ 7%&!'!<! #"+,+"#+%+$ "+%+&#+ ! ,&-'!'##"!" (+ ! )!'- "+ 6 !9-$ "-$ "+ +-$$#/! 4()#-!"-.

S-()#*

S+! ' V!&#!'+ !+!%-&#! )-%#(!F"(&!)#* "+ ( )-,-+%+ + !9-$ %#++

"#$%&#'()#* +3,-+)#! )- μ= '=4

S( "+$#"!" "+ ,&-'!'##"!" +$

f ( / )= 1

4e− /

4 , />0

L! ,&-'!'##"!" (+ ( )-,-+%+ $#/! 4()#-!"- ! )!'- "+ 6 !9-$:

P (0 ≥6)=1−∫0

6

1

4 e

− /

4

d/=0.2231



Ilustración 1.- /robabilidad de eemplo es i0ual rea pintada

S+! X' V!&#!'+ !+!%-&#! "#$)&+%!F)!%#"!" "+ )-,-+%+$ (+ $#/!4()#-!"- (+/- "+ 6 !9-$ %#++ "#$%&#'()#* '#-#! )- ?@ ! ? 0.22@1

S( 4()#* "+ "#$%&#'()#* + ,&-'!'##"!" +$:f ( x )=(n

x) p x (1− p)n− x=(3 x )0.02231 x

0.77693− x

E%-)+$

P ( x=2 )= f (2 )=(32)0.223120.7769

3−1=0.1160=11.6

DISTRIBUCION JI.CUADRADODe%iici&U! !&#!'+ !+!%-&#! )-%#(! 3 %#++ "#$%&#'()#* ICUADRADO $# $( "+$#"!" "+

,&-'!'##"!" +$%! "!" ,-& f ( x )={ 1

2

.

2 ) (. /2)

x

.

2−1

e− x/2

, x>0

0, para otro x

E$%! "#$%&#'()#* %#++ ( ,!&=+%&- : X 0 7 $+ "+-#! +&- "+ /&!"-$ "+ #'+&%!"



Ilustración 11- +r$,co de la distribución i-Cuadrado con v"

Medi" * V"ri"+" de l" Distribuci& JI.Cu"dr"do

S# +$ (! !&#!'+ !+!%-&#! )-%#(! )- "#$%&#'()#* #C(!"&!"- +%-)+$

Medi": μ= E ( x )=.

V"ri"+": σ 2=. ( x )=2.

DISTRIBUCIÓN BETADe%iici&

U! !&#!'+ !+!%-&#! )-%#(! 3 %#++ "#$%&#'()#* '+%! $# $( "+$#"!" "+

,&-'!'##"!" +$%! "!" ,-& f ( x )={ ) (( + ')

) (( ) ) ( ' ) x

( −1(1− x) '−1,0< x<1

0, para otro x

E "-"+ [X0 YX0 $- -$ ,!&=+%&-$ ,!&! +$%+ -"+-. ) () +$ ! 4()#*

G!!.Medi" #"ri"+" !"r" L" Distribuci& Bet"

S# X +$ (! !&#!'+ !+!%-&#! )- "#$%&#'()#* '+%! +%-)+$

Medi"' μ= E ( x )= (

( + '

V"ri"+"' σ

2=V ( x )= ('

( ( + ' )2(( + '+1)

E-e,!lo'



U "#$%&#'(#"-& "+ )#+&%- ,&-"()%- +! ! ##)#- "+ )!"! $+!!. L! ,&-,-&)#* "+!&%)(- (+ +"+ $+!!+%+ $+ ,(+"+ -"+!& )- ! "#$%&#'()#* '+%! )- [? 7Y?2.

! E)(+%&+ ! ,&-'!'##"!" (+ + !/(! $+!! +"! ! +-$ 0' E)(+%&+ + !-& +$,+&!"- "+ ! ,&-,-&)#* "+ +%! $+!!

S-()#*

S+! X' P&-,-&)#* "+ !&%)(- (+ +"+ $+!!+%+ F!&#!'+ !+!%-&#!)-%#(! $( "+$#"!" "+ ,&-'!'##"!" +$

f ( x )= ) (4+2)

) (4) ) (2) x

4−1(1− x)2−1=20 x3 (1− x ),0< x<1

! P ( x ≥0.9)=20∫0.9

1

x3 (1− x )dx=0.82=8.2

E$ + =&+! !&)!"! + + $#/(#+%+ /&!4#)-

Ilustración 12- rea de probabilidad del literal a

' μ= E ( x )= (

( + '=

4

4+2=

2

3 F+"+ + ,&-+"#- 2H@ "+ ,&-"()%- )!"!

$+!!

DISTRIBUCIÓN DE EIBULLDe%iici&

U! !&#!'+ !+!%-&#! )-%#(! 3 %#++ "#$%&#'()#* "+ \+#'( $# $( "+$#"!" "+

,&-'!'##"!" +$%= "!"! ,-& f ( x )={(' x '−1

e−( x

'

, x>0

0, para otro x

Medi" * V"ri"+" !"r" L" Distribuci& de eibull

S# +$ (! !&#!'+ !+!%-&#! )-%#(! )- "#$%&#'()#* "+ \+#'( +%-)+$

Medi"' μ= E ( x )=( −1

' ) (1+1/ ')



V"ri"+"' σ 2=V ( x )=(

−2

' [ ) (1+ 2

' )−( ) (1+1/ '))2]

E-e,!loS(,*/!$+ (+ ! #"! %# + 8-&!$ "+ ( )-,-+%+ ++)%&*#)- %#++ "#$%&#'()#* "+

]+#'( )- [? 0.1 Y?0.5! C!)(+ ! #"! %# ,&-+"#-' C!)(+ ! ,&-'!'##"!" (+ "(&+ =$ "+ @00 8-&!$

S-()#*S+! X: V#"! %# + 8-&!$F!&#!'+ !+!%-&#! )-%#(! $( "+$#"!" "+,&-'!'##"!" :

f ( x )=(' x '−1

e−( x

'

=0.05 x−0.5

e−0.1 x

0.5

! μ=(

−1

' ) (1+ 1

' )=(0.1)−1

0.5 ) (1+ 1

0.5 )=0.1−2

) (3 )=200ora

' P( x>300)=∫300

#

0.05 x−0.5

e−0.1 x

0.5

dx

M+"#!%+ ! $($%#%()#* /= x0.5⇒d/=0.5 x

−0.5dx=0.5( 1 / )dx⇒dx=

/

0.5d/

S+ -'%#++

P( x>300)=0.05 ∫√ 300

#1

/ e

−0.1 / /

0.5d/=0.1 ∫

√ 300

#

e−0.1

d/

¿1− P( x ≤300)=1−0.1 ∫0

√ 300

e−0.1

d/=0.177



BIBLIO2RA$FA$.Rod,i4e- O?eda L. PROBABILIDAD ESTADÍSTICA B@SICA PARA IN!ENIEROS

!a;ail E+cela S8e,io, Politcnica del Lito,al '%%0.

'.3al8ole RG M;e,+ RG M;e,+ S. PROBABILIDAD ESTADÍSTICA PARA IN!ENIEROS.SeHta ed. Roi4 "=+e- PEG edito,. MeHico PRENTICE2&ALLG&ISPANOAMERICANAGS.A. $###.

distribuciÓn de probabilidades discretas y continua

Documents