Andrés Miniguano Trujillo
ESCUELA POLITECNICA NACIONAL FACULTAD DE CIENCIAS Materia: PROBABILIDADES Y ESTADÍSTICA Deber N. 3: Variables Aleatrias Dis!retas ". Si
X
es #$a %ariable %ariable aleatria aleatria !$ &#$!i'$ &#$!i'$ (e (istrib#! (istrib#!i'$ i'$ a!#)#lati a!#)#lati%a: %a:
F ( x )=
{
0 0.1 0.15 0.40 0.65 0.85 1
si x <0 si −4 ≤ x < 1 si 1 ≤ x < 6 si 6 ≤ x < 9 si 9 ≤ x < 14 si 14 ≤ x < 20 si x ≥ 20
*alle: a+ La &#$ &#$!i' !i'$ $ (e (e ,r ,rbab babili( ili(a( a( (e X .
p ( x x )=
{
x =−4 x =1 x =6 x =9 x =14 x =20
si si si si si si
0.10 0.05 0.25 0.25 0.20 0.15
b+ La Es,e Es,era$ ra$-a -a la la Va Varia$-a ria$-a (e (e X . E [ X ] = μ X =
∑ x p ( x x ) i
i
i
E [ X ] =(−4 ) ( 0.10 )+ 1 ( 0.05 ) + 6 ( 0.25 )+ 9 ( 0.25 ) + 14 ( 0.20 ) + 20 ( 0.15 ) E [ X ] =9.2 2 2 V ( ( X )= E [ X ]−( E [ X ] ) V ( ( X ) =16 ( 0.10 )+ 1 ( 0.05 ) + 36 ( 0.25 ) + 81 ( 0.25 ) + 196 ( 0.20 ) + 400 ( 0.15 )−9.2 V ( ( X ) =130.1− 84.64= 45.46
!+
P ( X > 9 ) P ( X > 9 ) =1− F ( 9 )= 1−0.65 =0.35
(+
P (3 < X < 13 ) P ( 3 < X < 13 ) = F ( ( 13 ) − F ( 3 )− p ( 13 ) =0.65 −0.15 − 0.25=0.25
e+
P ( X > 7| X ≥ 12 ) P ( X > 7| X ≥ 12 )=
&+
2
P ( X > 7 ∩ X ≥ 12 ) P ( X ≥ 12 ) = =1 P ( X ≥ 12 ) P ( X ≥ 12 )
P ( X > 8| X ≥ 4 ) P ( X > 8| X ≥ 4 )=
1 − F ( 8 ) P ( X > 8 ∩ X ≥ 4 ) P ( X > 8 ) 1−0.40 = = = =0.75 P ( X ≥ 4 ) P ( X ≥ 4 ) 1 − F ( 4 ) + p ( 4 ) 1− 0.15 −0.05
/. U$a %ari %ariaable ble alea aleat tri riaa X t)a t)a sl sl ls %alre %alress 01 21 a 1 !$ P ( X = 4 ) =0.5 1 P ( X =6 )=0.3 1 P ( X =a )= p . Si se sabe #e la es,era$-a (e X es i4#al a 51 6alle ls %alres (e a p .
Andrés Miniguano Trujillo
8 = E [ X ] = μ X =
∑ x p ( x ) i
i
i
=( 4 ) ( 0.5 )+ 6 ( 0.3 )+ ap 4.2 =ap 8
3. La &#$!i'$ (e ,rbabili(a( (e #$a %ariable aleatria (is!reta
p ( x )=
{
0.18 0.09 0.15 0.22
0.20 0.16 0
si si si si si si si
X
est7 (a(a ,r :
x =−18 x =−7 x =4 x =8 x =13 x =19 x es otrovalor
*alle: a+ La &#$!i'$ (e (istrib#!i'$ (e X .
F ( x )=
{
0 0.18 0.27 0.42 0.64 0.84 1
si x <−18 si −18 ≤ x <−7 −7 ≤ x < 4 si 4 ≤ x <8 si 8 ≤ x < 13 si si 13 ≤ x < 19 si x ≥ 19
b+ La ,rbabili(a( (el e%e$t A : X sea mayor o igual a -7 y menor que 12 . P (−7 ≤ X < 12 )= F (12 )− F (− 7 )− p ( 12 ) + p (−7 ) =0.64 − 0.27 −0.22 + 0.09= 0.24 P ( X ≥ 4|0 < X < 13 )
!+
P ( X ≥ 4|0 < X < 13 )=
P ( X ≥ 4 ∩ 0 < X < 13 ) P ( 4 ≤ X < 13 ) F ( 13 ) − F ( 4 ) − p (13 )+ p ( 4 ) 0.84 = = = 0. P ( 0 < X < 13 ) P ( 0 < X < 13 ) F ( 13 )− F ( 0 )− p ( 13 )
(+ La es,era$-a %aria$-a (e X . E [ X ] =(−18 ) ( 0.18 ) + (−7 ) ( 0.09 )+ 4 ( 0.15 ) + 8 ( 0.22 ) + 13 ( 0.20 ) + 19 ( 0.16 )=4.13 2 V ( X ) =324 ( 0.18 )+ 49 ( 0.09 ) + 16 ( 0.15 ) + 64 ( 0.22 )+ 169 ( 0.20 ) + 361 ( 0.16 ) −4.13 V ( X ) =170.77 −17.0569=153.7131
0. La &#$!i'$ (e ,rbabili(a( (e #$a %ariable aleatria (is!reta X est7 (a(a ,r:
p ( x )=
{
0.2 0.08 0.1 0.3 0.15 0.17
si si si si si si
x =−3 x =0 x =2 x =5 x =8 x =12
*alle: a+ La &#$!i'$ (e (istrib#!i'$ (e X .
Andrés Miniguano Trujillo
{
si x <−3 si −3 ≤ x < 0 0.2 0.28 si 0 ≤ x< 2 F ( x )= 0.38 si 2 ≤ x < 5 0.68 si 5 ≤ x <8 0.83 si 8 ≤ x < 12 1 si x ≥ 12 0
b+ La es,era$-a la %aria$-a (e X . E [ X ] =(−3 ) ( 0.2 ) + 0 + 2 ( 0.1 )+ 5 ( 0.3 )+ 8 ( 0.15 ) + 12 ( 0.17 )= 4.34 2 V ( X )=9 ( 0.2 ) + 0 + 4 ( 0.1 ) + 25 ( 0.3 ) + 64 ( 0.15 ) + 144 ( 0.17 )−4.34 V ( X )= 43.78−18.8356 =24.9444
!+
P (− 1 < X < 8 ) 1 P ( X ≥ 1| X > 1 ) P (−1 < X < 8 )= F ( 8 ) − F (− 1 ) − p ( 8 )= 0.83− 0.2 −0.15= 0.48 1 − F ( 1 ) =1 P ( X ≥ 1| X > 1 )= 1 − F ( 1 )
8. U$a #r$a !$tie$e 8 blas bla$!as 3 $e4ras. Si se e9trae$ 3 blas al )is) tie), X re,rese$ta: El $;)er (e blas bla$!as1 6alle: Redorridode X = { 0,1,2,3 }
:
¿=¿
5 C 3∗3 C 0 8 C 3
p (3 ) = P [ X =3 ]
¿
P (B ∩ B ∩ B)
¿
¿
x =2
:
p ( 2 )= P [ X =2 ]
=0.1786
a+ La &#$!i'$ (e ,rbabili(a( (e (istrib#!i'$ (e X .
{
0.1786
p ( x ) =
0.5357 0.2679 0.0178
{
0 0.1786
F ( x )= 0.7143 0.9822 1
si si si si
x =0 x =1 x =2 x =3
si x <0 si 0 ≤ x < 1 si 1 ≤ x < 2 si 2 ≤ x < 3 si x≥3
b+ La ,rbabili(a( (e #e al )e$s / blas sea$ bla$!as. P ( X ≥ 2 )=1 − F ( 2 ) + p ( 2 )=0.5535
!+ La ,rbabili(a( (e #e #$a (e las blas sea $e4ra P ( X =2 ) =0.5357
2. E$ !iert labratri se &re!e$ 0 ti,s (e e97)e$es: El ,ri)er (#ra /< )i$#ts es #tili-a( el 08= (e las %e!es. El se4#$( (#ra 3< )i$#ts es #tili-a( el /8= (e las %e!es. El ter!er (#ra 0< )i$#ts es #tili-a( el /<= (e las %e!es. El !#art (#ra 2< )i$#ts es #tili-a( el "<= (e las %e!es.
¿ P [ ( BBN ) ∪ ( BN
Andrés Miniguano Trujillo
El !st (e ls e97)e$es est7 (a( ,r C ( X ) =10.000−3 X + 2 X 2 >!e$ta%s (e ('lar+1 ($(e X re,rese$ta el $;)er (e )i$#ts e),lea(s e$ el e9a)e$. p ( x ) =
{
F ( x )=
x =20 x =30 x =40 x =60
0.45 0.25 0.20 0.10
si si si si
{
si x < 20 si 20≤ x < 30 si 30 ≤ x < 40 si 40 ≤ x < 60 si x ≥ 60
0 0.45 0.70 0.90 1
a+ *alle la ,rbabili(a( #e ,r #$ e9a)e$ se (eba ,a4ar e$tre /<.<<< 8<.<<< !e$ta%s. 2
20000 =10000 −3 X + 2 X ⇒ X =71.4646 2 50000 =10000 −3 X + 2 X ⇒ X =142.1733 P ( 71.4646 ≤ X ≤ 142.1733 )= F ( 142.1733 )− F (71.4646 )+ p ( 71.4646 ) =0
b+ *alle el !st es,era( la %aria$-a (el !st. E [ X ] =20 ( 0.45 ) + 30 ( 0.25 ) + 40 ( 0.20 )+ 60 ( 0.10 )=30.5 2 C ( E [ X ])= 10.000 −3 ( 30.5 ) + 2 ( 30.5 ) =11769 [ centavos ] V ( X )= 400 ( 0.45 )+ 900 ( 0.25 ) + 1600 ( 0.20 ) + 3600 ( 0.10 )− 30.5 =154.75 2 C ( V ( X ) ) =10.000 −3 ( 154.75 ) + 2 ( 154.75 ) =57430.88 [ centavos ] 2
?. U$a #r$a !$tie$e 5 blas $e4ras 2 bla$!as. Si se %a$ a sa!ar 0 blas la %ariable aleatria X es: N;)er (e blas bla$!as bte$i(as1 6alle la &#$!i'$ (e ,rbabili(a( la &#$!i'$ (e (istrib#!i'$ (e si: X
a+ Se sa!a$ las 0 blas !$ re,si!i'$. Redorridode X = { 0,1,2,3,4 }
:
p ( 4 ) = P [ X = 4 ]
¿
P (B ∩ B ∩ B ∩ B)
¿
6 4∗8 C 0 ¿=¿ C =0.015
¿
x= 3
:
p ( 3 ) = P [ X =3 ]
14 C 4
{
0.0699 0.3357
p ( x ) = 0.4196 0.1598 0.015
si si si si si
x =0 x =1 x= 2 x =3 x =4
¿ P [ ( B
Andrés Miniguano Trujillo
F ( x )=
{
0 0.0699 0.4056 0.8252 0.985 1
si x< 0 si 0 ≤ x < 1 si 1 ≤ x < 2 si 2≤ x < 3 si 3 ≤ x < 4 si x≥ 4
b+ Se sa!a$ las 0 blas si$ re,si!i'$. x = 4 :
p ( 4 )= P [ X = 4 ] ¿
x = 3 :
p ( 3 )= P [ X =3 ]
¿
x = 2 :
p ( 2 )= P [ X =2 ]
¿
x = 1 :
p ( 1 )= P [ X =1 ]
¿
x = 0 :
p ( 0 )= P [ X = 0 ]
¿
{
0.1799 0.0337
si si si si si
x =0 x =1 x= 2 x =3 x =4
{
si si si si si si
x <0 0 ≤ x <1 1 ≤ x <2 2 ≤ x <3 3 ≤ x <4 x ≥ 4
0.1066 0.3199
p ( x ) = 0.2399
F ( x )=
0 0.1066 0.4265 0.6664 0.8463 1
6 6
6 6
= 0.0337 14 14 14 14 8 6 6 6 =0.1799 4 14 14 14 14 8 8 6 6 =0.2399 4 14 14 14 14 8 8 8 6 =0.3199 4 14 14 14 14 8 8 8 8 = 0.1066 14 14 14 14
5. La ,rbabili(a( (e #e el al#)$ A a,r#ebe #$ e9a)e$ es <.2 (e #e el al#)$ B l a,r#ebe es <.5. Si la %ariable aleatria X es: N;)er (e al#)$s #e a,r#eba$ el e9a)e$ e$tre A B 1 6alle: Redorridode X = { 0,1,2 } x =0
¿
:
p ( 0 )= P [ X = 0 ]
¿=¿ 1 −0.6 −0.8− 0.48=0.08
¿
c c ¿ ¿ P ( A ∩B ) ¿ P ( A +B )¿ ¿ ¿ ¿ : x =1 p ( 1 )= P [ X =1 ]
a+ Las &#$!i$es (e ,rbabili(a( (e (istrib#!i'$ (e X .
Andrés Miniguano Trujillo
E [ X ] =2 ( 0.025 ) + 3 ( 0.0875 ) + 4 ( 0.175 ) + 5 ( 0.2875 )+ 6 ( 0.425 )=5 V ( X )= 4 ( 0.025 ) + 9 ( 0.0875 ) + 16 ( 0.175 ) + 25 ( 0.2875 ) + 36 ( 0.425 ) −25= 1.175
"". Ciert )(i! atie$(e 3 ,a!ie$tes el /5= (e ls (as 0 ,a!ie$tes el //= (e ls (as 8 ,a!ie$tes el "?= (e ls (as 2 ,a!ie$tes el "8= (e ls (as ? ,a!ie$tes el "<= (e ls (as 5 ,a!ie$tes el 5= (e ls (as. El )(i! !bra "8 ,r !$s#lta ls !sts (iaris (el !$s#ltri s$ i4#ales a 8<. *alle:
p ( x )=
{
0.28 0.22 0.17 0.15 0.10 0.08
si si si si si si
x =3 x =4 x =5 x =6 x =7 x=8
a+ La &#$!i'$ (e (istrib#!i'$ (e X : G N;)er (e ,a!ie$tes #e el )(i! atie$(e ,r (aH
{
si si si F ( x )= si 0.82 si 0.92 si 1 si 0 0.28 0.50 0.67
x <3 3 ≤ x< 4 4 ≤ x <5 5 ≤ x <6 6 ≤ x <7 7 ≤ x <8 x ≥8
b+ La ,rbabili(a( (el e%e$t A : GEl )(i! atie$(a e$ #$ (a )$i) 3 ,a!ie$tes )e$s (e 2 ,a!ie$tesH. !+
P ( 3 ≤ X < 6 )= F ( 6 )− F ( 3 ) − p ( 6 ) + p ( 3 )=0.67 P ( X ≥ 5|3 < X < 7 ) P ( 5 ≤ X < 7 ) F ( 7 )− F ( 5 ) − p ( 7 ) + p ( 5 ) = =0.5926 P ( X ≥ 5|3 < X < 7 ) = P ( 3 < X < 7 ) F ( 7 )− F ( 3 ) − p ( 7 )
(+ Las #tili(a(es (iarias es,era(as. E [ X ] =3 ( 0.28 ) + 4 ( 0.22 ) + 5 ( 0.17 ) + 6 ( 0.15 ) + 7 ( 0.10 )+ 8 ( 0.08 ) = 4.81 U = 4.81 ( 15 )−50= $ 22.15 e+ La %aria$-a (e X . V ( X ) =25.71 −23.1361= 2.5739
"/. El labratri GB#e$a Sal#(H reali-a "8 e97)e$es "5= (e ls (as /< e97)e$es el //= (e ls (as /8 e97)e$es el /3= (e ls (as 3< e97)e$es el /<= (e ls (as 38 e97)e$es el "<= (e ls (as 0< e97)e$es el ?= (e ls (as.
p ( x ) =
{
0.18 0.22 0.23 0.20 0.10 0.07
si si si si si si
x =15 x =20 x =25 x =30 x =35 x =40
Cada examen cuesta 25 dólares y los costos diarios por los exámenes realizados son iguales a 300 dólares !alle"
a+ La &#$!i'$ (e (istrib#!i'$ (e X : G N;)er (e e97)e$es #e el labratri reali-a (iaria)e$teH.
Andrés Miniguano Trujillo
F ( x )=
{
0 0.18 0.40 0.63 0.83 0.93 1
si si si si si si si
x < 15 15 ≤ x < 20 20 ≤ x < 25 25 ≤ x < 30 30 ≤ x < 35 35 ≤ x < 40 x ≥ 40
b+ La ,rbabili(a( (el e%e$t A : GEl labratri reali-a )as (e /< e97)e$es )e$s (e 35 e97)e$esH. P ( 20 < X < 38 ) = F ( 38 )− F ( 20 )− p ( 38 )=0.53
!+ Las #tili(a(es (iarias es,era(as. E [ X ] =15 ( 0.18 ) + 20 ( 0.22 ) + 25 ( 0.23 ) + 30 ( 0.20 ) + 35 ( 0.10 ) + 40 ( 0.07 )=25.15 U =25.15 ( 25 ) −300= $ 328.75 (+ La %aria$-a (e X . 2 V ( X ) =225 ( 0.18 ) + 400 ( 0.22 )+ 625 ( 0.23 ) + 900 ( 0.20 ) + 1225 ( 0.10 ) + 1600 ( 0.07 )− 25.15 =686.
"3. E$ #$ labratri !l$i! se sabe #e la (e)a$(a (iaria (e !iert ti, (e e9a)e$ tie$e el si4#ie$te !),rta)ie$t: De)a$(a: < " / 3 Prbabili(a(: <." <./8 <.3 <.38 Si el !st ,r e9a)e$ es (e 8 ('lares1 !al!#le el i$4res (iari #e se es,era bte$er !$ ese ti, (e e9a)e$ la %aria$-a =5 E [ X ] =5 [ 0 + 0.25 + 0.6 + 1.05 ] = $ 9.5 V ( X )=0 + 0.25 + 1.2 + 3.15− 3.61 =0.99
"0. La E),resa COMPUUEJOS S.A. est7 i$teresa(a e$ ,r)!i$ar el K#e4 ,ares i),ares #e !$siste e$ l si4#ie$te: Para !a(a i$te$t1 #$a !),#ta(ra ,r,r!i$a al a-ar tres $;)ers: El ,ri)er $;)er ,#e(e ser el "1 el / el 3 El se4#$( $;)er ,#e(e ser el 0 el 8 El ter!er $;)er ,#e(e ser el 21 el ? el 5 La %ariable X es: La s#)a (e ls tres $;)ers bte$i(s e$ el i$te$t A$tes (e i$i!iar el K#e41 #$ (e ls (s K#4a(res es!4e la ,!i'$ X ,ar el tr X i),ar. a+ La E),resa le !$s#lta a #ste( si el K#e4 es le4al es (e!ir1 si ls (s K#4a(res tie$e$ la )is)a ,rbabili(a( (e 4a$ar (es,#s (e )#!6s i$te$ts. Recorridode X = { 11,12,13,14,15,16 } x = 11: p ( 11 )= P [ X =11 ] = P ( 1,4,6 )=
111 323
=0.0556
x = 12: p ( 12 )= P [ X =12 ] = P [ (1,4,7 ) ∪ ( 1,5,6 ) ∪ ( 2,4,6 ) ]= 3
111 323
=0.1667
x = 13 : p ( 13 ) = P [ X =13 ]= P [ ( 1,4,8 ) ∪ (1,5,7 ) ∪ ( 2,4,7 ) ∪ ( 2,5,6 ) ∪ ( 3,4,6 ) ] = 5
1 1 1
=0.2778 323 111 = 0.2778 x = 14 : p ( 14 ) = P [ X =14 ]= P [( 1,5,8 ) ∪ ( 2,4,8 ) ∪ ( 2,5,7 ) ∪ ( 3,4,7 ) ∪ ( 3,5,6 )]= 5 323 1 1 1 =0.1667 x = 15 : p ( 15 ) = P [ X =15 ]= P [( 2,5,8 ) ∪ ( 3,4,8 ) ∪ ( 3,5,7 ) ]=3 323 111 =0.0556 x = 16 : p (16 ) = P [ X =16 ] = P (3,5,8 )= 323
