distribuciÓn de probabilidades discretas y continua
DESCRIPTION
DISTRIBUCIÓN DE PROBABILIDADES DISCRETAS Y CONTINUATRANSCRIPT
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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
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INTRODUCCIÓN
U! "#$%&#'()#* "+ ,&-'!'##"!" #"#)! %-"! ! /!! "+ !-&+$ (+ ,(+"+&+,&+$+%!&$+ )-- &+$(%!"- "+ ( +3,+&#+%-. U! "#$%&#'()#* "+ ,&-'!'##"!" +$$##!& ! "#$%&#'()#* "+ 4&+)(+)#!$ &+!%#!$ .S# +'!&/- + + "+ "+$)&#'#& +,!$!"- "+$)&#'+ ! ,&-'!'##"!" (+ ( ++%- $+ &+!#)+ + + 4(%(&- )-$%#%(7+ (!8+&&!#+%! 4("!+%! ,!&! ! ,&-$,+)%#! ,(+$%- (+ $+ ,(+"+ "#$+9!& (+$)+!&#- "+ !)-%+)##+%-$ 4(%(&-$ )-$#"+&!"- !$ %+"+)#!$ !)%(!+$ "+ "#+&$-$4+*+-$ !%(&!+$.
L!$ "+)#$#-+$ +$%!"$%#)!$ '!$!"!$ + ! +$%!"$%#)! #4+&+)#! $- 4("!+%!+$+ ! #+$%#/!)#* (+ $- +!(!"!$ + %;&#-$ "+ "#$%&#'()#* "+ ,&-'!'##"!"+$.
E + ,&+$+%+ %&!'!<- $+ +$%("#! "+ !+&! =/# -$ "#+&$- %#,-$ "+ "#$%&#'()#*,&-'!'#$%#)! )!&!)%+&#!&+-$ )!"! "#$%&#'()#* ! 4("!+%!)#* !%+=%#)! "+-$ "#+&$-$ &+$(%!"-$ - $+ +4-)!&! + + ,&+$+%+ %&!'!<-> $*- + ##%!&; !+$%("#- "+$)&#,%#- "+ ! "#$%&#'()#* "+ ,&-'!'##"!"+$ "#$)&+%!$.
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OBJETIVOS
• I"+%#4#)!& -$ )-)+,%-$ "+ 4()#* "+ ,&-'!'##"!" 7 4()#* "+ "+$#"!"
,!&! !&#!'+$ !+!%-&#!$ "#$)&+%!$ 7 )-%#(!$ &+$,+)%#!+%+.
• R+)--)+& ! "#$%&#'()#* '#-#! 7 + )-%+3%- "+ $( !,#)!)#*.
• C!)(!& ,&-'!'##"!"+$ ,!&! !&#!'+$ !+!%-&#!$ '#-#!+$.
• R+)--)+& ! "#$%&#'()#* -&! 7 + )-%+3%- "+ $( !,#)!)#*.
• C!)(!& ,&-'!'##"!"+$ ,!&! !&#!'+$ !+!%-&#!$ -&!+$ +"#!%+
%#,#4#)!)#* 7 ($- "+ %!'!$.
• I"+%#4#)!& )(="- +$ !)+,%!'+ !,&-3#!& (! "#$%&#'()#* '#-#! ,-& (!
-&! 7 )*- 8!)+&-.
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Tabla de contenidoDISTRIBUCIONES DE PROBABILIDAD DISCRETAS........................................................5
DISTRIBUCIÓN DISCRETA UNIFORME.......................................................................5
DISTRIBUCION BINOMIAL.........................................................................................DISTRIBUCIÓN BINOMIAL NE!ATI"A........................................................................#
DITRIBUCION !EOMETRICA................................................................................... $%
DISTRIBUCION &IPER!EOMETRICA........................................................................$$
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-"+-$ !%+=%#)-$ ,!&! )!)(!& ! ,&-'!'##"!" + !/(-$ ,&-'+!$ %,#)-$+ -$ (+ #%+&#++ !&#!'+$ !+!%-&#!$ "#$)&+%!$.
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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+$,(+$%!
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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%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ +$
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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 )
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¿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
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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'
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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
,
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V"ri"+"' σ 2=V [ X ]= 1
p(1
p−1)
E-e,!lo'
C!)(+ ! ,&-'!'##"!" (+ + + (#%- !!#+%- "+ %&+$ -+"!$ $+ -'%+/! %&+$$+-$ ,-& ,&#+&! +.
R+$,(+$%!
E + +3,+&#+%- "+ !!& %&+$ -+"!$ 8!7 &+$(%!"-$ ,-$#'+$.
E )!"! +$!7- ! ,&-'!'##"!" (+ $!/! %&+$ $+-$ +$ )-$%!%+ + #/(! ! 1H7 ! ,&-'!'##"!" (+ - $!/! %&+$ $+-$ +$ H.
E$%-$ +$!7-$ $- #"+,+"#+%+$ 7 ,-& ! ,&+/(%! )-)(#-$ (+ ! !&#!'+"+ #%+&;$ X %#++ "#$%&#'()#* /+-;%&#)! )- !)5678
S+! X' )!%#"!" "+ +$!7-$ 8!$%! -'%++& + ,&#+& “;3#%-” F!&#!'+ !+!%-&#!"#$)&+%!
?1 2@
P ( X = x )= f ( x )=( 18 )( 78 ) x−1
, x=1,2,3,…
P-& - %!%-
P ( X =5 )=f (5)=(18 )(
78 )
5−1
=0.0733
DISTRIBUCION 9IPER2EOMETRICA
De%iici&'
S+!
N' C!%#"!" "+ +++%-$ "+ )-<(%- "+ (+ $+ %-! ! (+$%&!
:: C!%#"!" "+ +++%-$ +3#$%+%+$ (+ $+ )-$#"+&! “;3#%-$”
N' T!!9- "+ ! (+$%&!
X: V!&#!'+ !+!%-&#! "#$)&+%! F+$ ! )!%#"!" "+ &+$(%!"-$ )-$#"+&!"-$ “;3#%-$”(+ $+ -'%#++ + ! (+$%&!
(?0 12 F$- -$ !-&+$ (+ ,(+"+ %-!& X
E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" "+ X +$
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f ( x )=(k
x )( N −k
n− x )( N
n ) , x=0,1,2,… ,n
Medi" * V"ri"+" de l" Distribuci& 9i!er4eo,3tric"
Medi": μ= E [ X ]=n K
N ,
V"ri"+": σ 2=V [ X ]=nK
N (1− K
N )( N −n
N −1 )
E-e,!lo'
U! )!<! )-%#++ '!%+&!$ "+ !$ )(!+$ +$%= + '(+ +$%!"- 7 !$ &+$%!%+$"+4+)%(-$!$. S+ %-! (! (+$%&! +#/#+"- ! !!& %&+$ '!%+&!$. C!)(+ !,&-'!'##"!" (+ + ! (+$%&! $+ -'%+/!.
! N#/(! '!%+&! + '(+ +$%!"-' A +-$ (! '!%+&! + '(+ +$%!"-) C!)(!& ! +"#! 7 !&#!!
R+$,(+$%!E$%+ +$ ( +3,+&#+%- "+ (+$%&+- $# &++,!- ,-& - %!%- +$ ( +3,+&#+%-
8#,+&/+-;%&#)- )-N); <T-%! "+ +++%-$ "+ )-<(%-=:)> <T-%! "+ +++%-$ )-$#"+&!"-$ “;3#%-$”=)? <%!!9- "+ ! (+$%&!=X' )!%#"!" "+ '!)%+&#!$ + '(+ +$%!"- + ! (+$%&! F!&#!'+ !+!%-&#!
E%-)+$ ! "#$%&#'()#* "+ ,&-'!'##"!" +$
f ( x )=( K
x )( N − K
n− x )( N
n ) =
(4 x)(9−4
3− x )(93)
, x=0,1,2,3
! P ( x=0 )= f (0 )=(40)(
9−4
3−0)(93)
=0.119
' P ( x ≥1 )=1− P ( x<1 )=1− f (0 )=1−0.119=0.881
) μ=3 (4
9 )=1.333
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σ 2=
3 (4 )9 (1−4
9 )( 9−3
9−1 )=0.555
DISTRIBUCION POISSONDe%iici&'
S+! X: V!&#!'+ !+!%-&#! "#$)&+%! )- "#$%&#'()#* "+ P-#$$- 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-#$$- "+%+&#+ ! ,&-'!'##"!"
(+:! E )(!(#+& 8-&! -)(&&! $-!+%+ 1 +&&-& ' E )(!(#+& 8-&! -)(&&! ! +-$ @ +&&-&+$
R+$,(+$%!S+! X' V!&#!'+ !+!%-&#! "#$)&+%!F)!%#"!" "+ +&&-&+$ ,-& 8-&! ?5 F,&-+"#-"+ +&&-&+$ "+ %&!$#$#* + 1 8-&!
"= P ( x=1 )=f (1 )= e−55
1
1! =0.0337
b= P ( x ≥3 )=1− P ( x ≤2 )=1−(f (0 )+ f (1 )+ f (2 ) )=1−0.1247=0.8743
DISTRIBUCIÓN CONTINUA
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S- !(+!$ + !$ (+ ! 4()#* "+ "#$%&#'()#* +$ (! 4()#* )-%#(!. S+)-&&+$,-"+ )- + ,&#+& %#,- "+ /&=4#)! #$%-.
G++&!+%+ $+ )-&&+$,-"+ )- !&#!'+$ !$-)#!"!$ ! +3,+&#+%-$ + -$ )(!+$! !&#!'+ +"#"! ,(+"+ %-!& )(!(#+& !-& + ( #%+&!-> +"#)#-+$ '#-;%&#)!$,-& +<+,-.
U )!$- ,!&%#)(!& "+%&- "+ !$ !&#!'+$ !+!%-&#!$ )-%#!$ 7 ! )(! ,+&%++)+%-"-$ -$ +<+,-$ ($(!+%+ (%##!"-$ $- !$ "+-#!"!$ !&#!'+$ !+!%-&#!$!'$-(%!+%+ )-%#(!$.
DISTRIBUCIÓN UNI$ORME CONTINUADe%iici&'
S+! : V!&#!'+ !+!%-&#! )-%#(! %#++ "#$%&#'()#* +$%= "!"! ,-&
f ( x )=
{
1
b−a, a ≤ x ≤ b
0 , paraotro x
A' $- -$ ,!&=+%&-$ ,!&! +$%+ -"+-
R+,&+$+%!)#* /&=4#)! "+ ! "#$%&#'()#* (#4-&+ )-%#(!
Ilustración 3- Distribución Continua Uniforme
Medi" * V"ri"+" de l" Distribuci& Ui%or,e Cotiu"Medi": μ= E ( x )= 1
2(a+b)
V"ri"+": σ 2=V ( x )=
1
12(b−a)2
E-e,!lo'C(!"- 4!! )#+&%! )-,-+%+ "+ (! =(#! +$%! "+'+ "+%++&$+ 8!$%! (+ $+!&+,!&!"-. S(,-#+"- (+ + %#+,- "+ &+,!&!)#* ,(+"+ %-!& )(!(#+& !-& +%&+ 17 5.
"= C"lcule l" !rob"bilid"d @ue l" dur"ci& to,e "l ,eos or"s8
S-()#*
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: !&#!'+ !+!%-&#! )-%#(! F"(&!)#* "+ ! &+,!&!)#* %#++ "#$%&#'()#*(#4-&+ ,-& - %!%- $( 4()#* "+ "+$#"!" +$
f ( x )= 1
b−a=
1
5−1=
1
4,1≤ x ≤5
P ( x ≥2 )=∫2
5
14
dx=34=75
b= C"lcule el #"lor es!er"do de l" dur"ci& de l" re"!"rici&
E ( x )=1
2(a+b )=
1
2(1+5 )=3ora!
DISTRIBUCION CONTINUA NORMALDe%iici&:
S+! X' V!&#!'+ !+!%-&#! )-%#(! )- +"#! 7 !&#!! σ 2 X %#++
"#$%&#'()#* -&! $# $( 4()#* "+ "+$#"!" +$:
f ( x )= 1
σ √ 2" e
−1
2( x− μ
σ )2
,−#< x<+#
L! /&!4#)! "+ 4 +$ $##!& ! ,+&4# "+ )-&%+ +&%#)! "+ (! )!,!! 7 %#++ !$$#/(#+%+$ )!&!)%+&$%#)!$:
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
,−#< $<+#
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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
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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-()#*
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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'
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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 +$ + =&+! &+$!%!"! + + /&!4#)-
P ( X >8)=1 P ( X ≤ )=1 1
16
∫0
8
x2
e
− x
2 dx
P!&! #/&+$!& $+ ,(+"+ !,#)!& "-$ +)+$ ! %;)#)! "+ #%+/&!)#* ,-& ,!&%+$:
∫ x2
e
− x
2
"3
-= x 2⇒d-=2 x dx
d.=e− x
2 dx ⇒.=−2e− x
2
¿−2 x2 e
− x
2 +4∫ x e
− x
2 dx
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∫ x e− x
2 dx
-= x ⇒d-=dx
d.=e− x
2dx
⇒ .=−2e− x
2
¿−2 x e− x2 +2∫e
− x2 dx
S($%#%(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
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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 $+ "+-#! +&- "+ /&!"-$ "+ #'+&%!"
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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'
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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/ ')
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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
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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. $###.