“UNIVERSIDAD DE LAS FUERZAS ARMADAS - ESPE” EXTENSIÓN EXTENSI ÓN - LAT LATACUNGA ACUNGA
ELECTRÓNICA E INSTRUMENTACIÓN
PROCESOS ESTOCASTIC ESTOCASTICOS OS CAP.. 5 PROBABILIDADES DE SCHAUM CAP SCH AUM
NOMBRE: Men M!"#$#% NIVEL: &UINTO
CAPITULO 5 VARIABLES ALEATORIAS ALEATORIAS Y VALOR ESPERADO
5.5'.-S!(%n) *!e !n +"#,e e%"# X %/ %0 +%"e0 -12 32 '2 4 $%n 0 ("%,,##e0 "e0(e$#+0. k + 2 2 k −3 3 k −4 k + 1 , , , 10
10
10
10
En$%n"" #0"#,!$#6n 7 e +%" "e0(e$#+% e X. k + 2 10
+
−3
2 k
10
+
−4 k +1 + =1
3 k
10
10
k + 2 + 2 k −3 + 3 k + 4 + k + 1=10
k =2 k + 2 10
=0.4
−3
2 k
10
−4
3 k
10
k + 1 10
=0.1
=0.2
=0.3 x P(X=x)
4 0.4
2 0.1
3 0.2
7 0.3
E ( x )=(−4 )∗( 0.4 ) + ( 2 )∗( 0.1 ) + ( 3 )∗( 0.2 ) + ( 7 )∗(3 ) E ( x )=1.3 5.51.- Se n8 !n (" e %0. Se 9 e /n#/% e %0
%0 n;/e"%0 *!e %$!""en. En$!en"e #0"#,!$#6n 7 e +%" e0(e"% e 9. x f(x)
1 11/36
2 9/36
3 7/36
4 5/36
5 3/36
6 1/36
E ( x ) =1
E ( x )=
( ) ( ) ( ) ( ) ( ) ( ) 11 36
91
+2
9
+3
36
7
36
+4
5
36
+5
3
36
+6
1
36
=2.53
36
5.55.- E (e0% (e0% e !n !n /%ne /%ne e*!# e*!## #," ," 1 +e$e0 e$e0.. Se Se < 0e$!en$# /=0 ") e $"0 *!e 0). En$!en"e #0"#,!$#6n 7 e +%" e0(e"% e <. >C%/("e $%n +"#,e e%"# X en e ("%,e/ 5.33?.
x f(x)
0 1/16
1 7/16
2 5/16
( ∗ )+( ∗ )+( ∗ )+( ∗ )+( ∗ )=
E ( x )=
0 1
1
16
7
2
16
5
16
3 2 16
4
1
16
27 16
3 2/16
4 1/ 1/16
=1.68
5.5@
[email protected] E (e0% e !n /%ne e0 e"% e /ne" *!e P ( H )=¾ yP ( T )= 1 / 4 2 0e n8 ' +e$e0. Se 9 e n;/e"% e $"0 *!e ("e$e. En$! En$!en en" "e e #0 #0" "#, #,!$ !$#6 #6n n e 9. 9. , En$ En$!en !en"e "e E>9? E>9?.. x f(x) a) ¼
∗1
4
∗1 ∗¿
4
64
( ) 1
4
3
1
∗1
∗3
4
4
=
9 64
1 1/64
2 9/64
3 27/64
( ) 3
4
3
¾
∗3
4
∗1
4
=
4
∗3 =
4
∗3
27 64
27 64
b) E ( x )=0
E ( x )=
( ) ( ) ( ) ( ) 1
64
+1
9
64
+2
27 64
27
+3
64
144 64
E ( x )=2.25
P ( H H ) =
5.54.- EL (e0% e !n /%ne e0 e"% e /ne" *!e 7
P (T ) =
1 3
2 3
. L /%ne 0e n8 0 *!e ("e8$ !n $" % 5
0e%0 0e%0.. En$!en En$!en"e "e e n! n!/e" /e"% % e0(e" e0(e"% % E e n8/ n8/#en #en%0 %0 e /%ne. H ,TH , TH , TTH , TTTH , TTTTH , TTTTT X ( ( H )=1 X ( TH )=2 X ( ( TTH ) =3 X ( ( TTTH ) =4 X ( TTTTH )=5 X ( TTTTT )=5
( )( )
1
P (1 ) = P ( H )= P (2 )= P (TH ) = 3
P ( 4 )= P (TTTH )=
2
1
3
3
( )( )( )( ) 2
2
2
1
3
3
3
3
P (5 )= P ( { TTTTH TTTTH , TTTTT TTTTT } )=
=
( )( )( )
2
= P ( 3 )= P ( TTH )= 9
2
2
1
3
3
3
=
8 81
[( )( )( )( )( )] [( )( )( )( )( )] 2
2
2
2
1
3
3
3
3
3
+
2
2
2
2
2
3
3
3
3
3
4 27
P (5 )= P ( { TTTTH TTTTH , TTTTT TTTTT } )=
E= E ( x )= 1
16 243
+
32 243
=
48 243
( )+ ( )+ ( )+ ( )+ ( ) 1
2
2
3
9
1
4
12
3
9
27
E= E ( x )= + +
+
4
3
32 81
4
27
+
8
81
5
48
243
240 243
E ( x )=2.6 .
5.5.- L ("%,,## e *!e e e*!#(% A )ne $!*!#e" !e)% e0 3. S!(%n) *!e A !e) $%n" B en !n %"ne%. E ("#/e" e*!#(% en )n" 3 !e)%0 0e)!#%0 % ' !e)%0 )n e %"ne%. En$!en"e e n!/e"% e0(e"% E e !e)%0 en e %"ne%. x f(x)
E ( x )=2
E ( x )=
2 2/4
3 2/8
4 2/16
5 4/32
() () ( ) ( ) 2 4
2
+3
8
+4
2
16
+5
4
32
23 8
E ( x )=2.9
5.5.- Un Un $ $ $%n# $%n#ene ene "n0# "n0#0% 0%"e0 "e022 e %0 $!e0 $!e0 3 e0=n e0=n ee$!%0%0. Se 0ee$$#%n !n "n0#0%" e $ 7 0e ("!e, 0 0 0ee 0ee$$ $$#% #%n n"" !n !n% % n% ee ee$ $!% !%0% 0%.. En$! En$!en en" "e e e n; n;/e /e"% "% e0(e"% E e "n0#0%"e0 *!e e,en e0$%)e"0e. P ( D D )= P ( B ) =
2 10 8
10
x f(x)
E ( x )=1
1 8/10
( ) ( ) ( ) 8
+2
10
16 90
+3
2
90
=
2 16/90
3 2/90
11 9
E ( x )=1.2
5.@.-Re0!e+ 5.@.-Re0!e+ e ("%,e/ ne"#%" (" e $0% en e $! ' e %0 "$!%0 0%n ee$!%0%0. P ( D D )=
P ( B ) =
3 10
7 10
x f(x)
E ( x )=1 (
E ( x )=
11 8
7 10
)+ 2 (
1 7/10 21 90
)+ 3 (
42 720
2 21/90
)+ 4 (
6 720
3 42/720
4 6/720
)
=1.4
5.@.- C#n$% $"0 $"0 e0=n n!/e"0 n!/e"0 e 5. Se 0$n 0$n %0 $"0 $"0 8" >0#n "e(%0#$#6n?. Se X 0!/ e %0 n;/e"%0 0ee$$#%n%0. En$! En$!en en" "e e #0 #0" "#, #,!$ !$#6 #6n n e X , En$!en"e "e E> E> 9 ?
a) x F(x)
3 0.1
4 0.1
5 0.2
U0n% e #)"/ e =",% 0e %,#ene 1
F ( 3 )=
5
4
1
F ( 4 )=
5
F ( 5 )=
5
F ( 6 ) =
5
1
F ( 7 ) =
5
+
+
5
5
+
+
5
+
10
1 10
5
+
5
+
5
+
+
+
∗1 4
5
5
=
∗1 4
1
∗1 4
5
1
∗1 4
1
1
∗1 4
1
∗1 4
1
1
∗1 4
1
=
∗1 4
5
=
∗1 4
1
∗1 4
∗1 4
1
∗1 4
5
1
∗1 4
1
+
∗1 4
1
1
∗1
∗1 4
=
1 5
1 5
=1 5
6 0.2
7 0.2
8 0.1
9 0.1
1
F ( 8 ) =
5
4
1
F ( 9 ) =
5
1
∗1 +
+
∗1 4
1
∗1 4
5
5
=
1 10
∗1 4
=
1 10
b) E ( x )=3 ( 0.1 )+ 4 ( 0.1 ) + 5 ( 0.2 ) + 6 ( 0.2 ) + 7 ( 0.2 ) + 8 ( 0.1 ) + 9 ( 0.1 ) E ( x )=6
[email protected] %e" $%n 5 ,%e%0 e !n ("e/#% e 2 ' ("e/#%0 e 5 $ !n%2 7 5 ("e/#%0 e 35 $ !n%. ? En$!en"e 0 )nn$#0 )nn$#0 e0(e"0 e !n !n ,%e. ,%e. ,? S# !n ,%e ,%e $!e0 JC!= e0 e +%" e0(e"% e !e)%K x f(x) f(x)
0 491/ 491/50 500 0
25 5/50 5/500 0
E ( x ) =0
50 3/50 3/500 0
100 1/50 1/500 0
( )+ ( )+ ( )+ ( ) 491 500
25
5
500
50
3
500
100
1
500 3
E ( x )= −1 4
)=−0.25 E ( x )=− 5.@'.- Un !)%" n8 ' /%ne0 e*!##,"0. E !)%" )n 5 0# %$!""en ' $"0. ' 0# %$!""en 3 $"0 7 0# 0%/ene %$!""e $". P%" %" ("e2 e !)%" (#e"e 5 0# %$!""en ' 0e%0. En$!en"e e +%" e !e)% (" e !)%". x f(x)
E ( x )=5
5 1/8
3 3/8
( ) ( ) ( ) ( )=− 1 8
+3
3
8
+1
1 8
−15
1 8
0.75
1-0.75=0.25
1 3/8
-15 1/8
[email protected] !)%" n8 %0 /%ne0 e*!##,"0. E !)%" )n ' 0# %$!""en 3 $"0 7 0# %$!""e !n $". P" *!e e !e)% 0e !0% JC!=n% JC!=n% e,e (e"e" e !)%" !)%" 0# n% %$!""e %$!""e n#n)!n $"K $"K
Para #=0 0
a
que
x f(x)
el
3 $
1 2/4
-a 1/4
jueg !ea ju!"
() () () =( ) ( ) =3
5 4
1
+1
4
2 4
– – a
1
4
4
a =5
MEDIA2 VARIANZA < DESVIACIÓN ESTNDAR
[email protected][email protected] En$!en En$!en"e "e /e# /e# e0=n" σ x
μ 2 +"# +"#n n8 8
2
σ x 2 7 e0+#$#6n
e $ #0"#,!$#6n. #0"#,!$#6n.
a X f(x)
2 $
3 1/2
8 1/4
)=(2 )( 1 / 4 )+( )+ ( 3 )( 1 / 2)+( 8 )( 1 / 4 ) E ( X )=( E ( x )=1 / 2 + 3 / 2 + 2
μ= E ( x )= 4 2
2
2
2
E ( X )=(2 )( 1 / 4 )+( 3 )( 1 / 2 )+( 8 )( 1 / 4 ) 2
E ( X )=1 + 9 / 2+ 16 2
E ( X )=21.5 2
2
Var ( x )= E ( X )− E ( 〖 x ) 〗
2
Var ( X )=21.5 −〖 ( 4 ) 〗 2
σ =Var ( X )=5.5 σ =√ ( 5.5 )=2.34
(b) x f(x)
-2 1/3
-1 1/2
7 1/6
E ( X )=(− )=(−2)( 1 /3 )+(−1 )( 1 / 2)+( )+ ( 7 )( 1 / 6 ) E ( x )=(−2 / 3 )+(− 1 / 2 )+( 7 / 6 )
μ= E ( x )= 0 2
2
2
2
E ( X )=(−2 )( 1 / 3 )+(−1 )( 1 / 2 )+( 7 )( 1 / 6 ) 2
E ( X )= 4 / 3 + 1 / 2 + 49 / 6 E ( X ) =10 2
x
¿ ¿
Var ( x )= E ( X )− E ¿ 2
2
Var ( X )=10 −( 0) 2
σ =Var ( X )=10 σ =√ 5.5 5.5 =3.2
5.@@.-En$!en"e /e# e !2 +"#n8 7 e0+#$#6n e0=n" e $ #0"#,!$#6n: #0"#,!$#6n:
x f(x)
-1 0.3
0 0.1
1 0.1
2 0.3
3 0.2
u= (− 1 ) ( 0.3 ) + ( 0 ) ( 0.1 ) + ( 1 ) ( 0.1 ) + ( 2 ) ( 0.3 ) + ( 3 ) ( 0.2 )
u=1
)+ ( 2) 2 ( 0.3 )+( 3 ) 2 ( 0.2 ) u ( x 2)=(−1) 2 ( 0.3 )+( 0 ) 2 ( 0.1 )+( 1 ) 2 ( 0.1)+( u ( x 2)= 3.4 var =3.4 – 1 =2.4 σ =√ 2.4 2.4 = 1.5
x f(x)
1 0.2
2 0.1
3 0.3
6 0.1
7 0.3
u=( 1 )( 0.2)+( )+ ( 2)( 0.1)+( )+ ( 3 )( 0.3 )+( 6 )( 0.1 )+( 7 )( 0.3 ) u= 4
u ( x 2)=(1 )2 ( 0.2 )+( 2) 2 ( 0.1 )+( 3 ) 2 ( 0.3)+( )+ ( 6 ) 2 ( 0.1)+( 8 )2 ( 0.3) u ( x 2)= 21.6
var =21.3 – 16 var =5.6 5.6 = 2.37 σ =√ 5.6
[email protected] Se X !n +"#,e e%"# $%n 0#)!#ene #0"#,!$#6n: #0"#,!$# 6n: x F(x)
En$!en"e En$!en"e /e# /e#
1 0.4
3 0.1
u 2 +"#n8
4 0.2
2
σ 2 7 e0+#$#6n e0=n"
e X. E ( x ) =1 ( 0.4 ) + 3 ( 0.1 ) + 4 ( 0.2 )+ 5 ( 0.3 ) E ( x )=3
5 0.3
E ( x )= 1 ( 0.4 ) + 9 ( 0.1 ) + 16 ( 0.2 ) + 25 ( 0.3 ) 2
2
E ( x )= 12 var ( x )= E ( x )− E ( x ) 2
2
var ( x ) =12− 9 var ( x )=3 σ =√ var var ( x ) σ =√ 3 =1.7
5.@ 5.@..-Se Sen n 9 +"# +"#, ,e e e e% %"# "# en e ("%, ("%,e e/ / ne ne"# "#%" %".. En$!en"e /e# +"#n8 7 e0+#$#6n e0=n" e $ +"#,e e%"#: a) % = 3x&2 b) % = x 2 ') % = 2x x
1 0. 0.4
3 0.1
4 0.2
5 0.3
%
5 0. 0.4
11 0.1
14 0.2
17 0.3
u ( y ) =( 5 ) ( 0.4 ) + ( 11 ) ( 0.1 ) +( 14 ) ( 0.2 ) + ( 17 ) ( 0.3 ) u ( y ) =11 5
¿ ¿ ¿ 2 ( 0.4 ) +(11) ¿ ¿ 2 ( 0.1 )+ ( 14 ) ¿ 17
u ( y ) =¿ 2
¿ ¿
u ( y )=148 2
var = 27 σ = √ 27 27 = 5.2
b) % F( F(%)
1 0.4
9 0.1
16 0.2
25 0.3
u ( y )=(1)( 0.4 )+( 9 )( 0.1 )+( 16 )( 0.2 )+( 25)( 0.3) u ( y )=12
u ( y 2)=(1 ) 2 ( 0.4 )+( 9) 2 ( 0.1 )+( 16 ) 2 ( 0.2 )+( 25 ) 2 ( 0.3 ) u ( y 2)= 247.2
var =247.2 – ( 12 ) 2 var =103.2 103.2 = 10.2 σ =√ 103.2
% F(%)
2 0.4
8 0.1
16 0.2
32 0.3
u ( y )=(2)( 0.4 )+( 8 )( 0.1 )+( 16 )( 0.2 )+( 32)( 0.3 ) u ( y )=14.4
u ( y 2)=(2 ) 2 ( 0.4 )+( 8) 2 ( 0.1 )+( 16 ) 2 ( 0.2 )+( 32 )2 ( 0.3) u ( y 2)= 366.4
var =366.4 – ( 14.4 ) 2 var =159.04 σ =√ 159.04 159.04 =12.6
5.@.- Se X !n +"#,e e%"# $%n 0#)!#ene #0"#,!$#6n: x f(x)
-1 0.2
1 0.5
2 0.3
En$!en"e /e# μ 2 +"#n8
2
σ
7 e0+#$#6n e0=n"
σ e X. E ( x )=(−1 ) ( 0.2 ) + ( 1 ) ( 0.5 ) + ( 2 ) ( 0.3 ) E ( x )=0.9 E ( x ) =(−1 ) 2
2
2
2
( 0.2 ) +( 1 ) ( 0.5 )+ ( 2 ) ( 0.3 )
E ( x ) =0.2 + ( 0.5 )+ ( 1.2 ) =1.9 2
var ( x )= E ( x x )−[ E ( x x ) ]
2
2
var ( x )=1.9 −(0.9 )
2
var ( x )=1.09 σ ( x ) =√ var var ( x ) σ ( x ) =1.04
5.4.- Se 9 +"#,e e%"# en e ("%,e/ 5.@. En$!en"e /e# +"#n8 7 e0+#$#6n e0=n" e $ +"#,e e%"# 7 >9? %ne 4
a)
ф ( x )= x
b)
ф ( x )= 3
')
ф ( x x ) =2 x −1
x
a) X F(x)
-1 0.2
1 0.5
2 0.3
F(%)
1 0.2
1 0.5
16 0.3
u ( y )=(1)( 0.2 )+( 1 )( 0.5 )+( 16 )( 0.3)
u ( y )=5.5
)+ (1) 2 (0.5 )+( 16 ) 2 ( 0.3 ) u ( y 2)=(1 ) 2 ( 0.2)+( u ( y 2)= 77.5
var =77.5 – ( 5.5) 2 var =47.25 47.25 = 6.87 σ =√ 47.25
b) F(%)
1/3 0.2
3 0.5
9 0.3
)+ ( 9 )( 0.3 ) u ( y )=(1 / 3)( 0.2)+( 3)( 0.5)+( u ( y )= 4.27
)+ ( 3 )2 ( 0.5)+( )+ ( 9 ) 2 ( 0.3) u ( y 2)=(1 / 3 ) 2 ( 0.2)+( u ( y 2)= 28.91
var =28.82 – ( 4.27 ) 2 var =10.59
σ =√ 10.59 =3.25 ') F(%)
1 0.2
2 0.5
8 0.3
)+ ( 8 )( 0.3 ) u ( y )=( 2)( 0.2)+( 3)( 0.5)+( u ( y )=3.6
)+ (3 ) 2 ( 0.5)+( )+ ( 8 ) 2 ( 0.3) u ( y 2)=( 2 ) 2 ( 0.2)+( u ( y 2)= 21.4 var =21.4 – ( 3.6 ) 2
var =8.44 σ =√ 8.44 8.44 = 2.91
5.4.5.4.- En$!en En$!en"e "e /e# /e# σ x
e0=n e0=n" "
μ 2 +"# +"#n n8 8
2
σ x 2 7 e0+#$#6n
e 0#)!#e 0#)!#ene ne #0"#,! #0"#,!$#6n $#6n e e %0 (!n%0 (!n%0 %ne %ne
p + q =1 .
x f(x)
a
b q
)=(a )( p )+( b)( q ) E ( X )=( )=( ap )+( bq ) E ( x )=( μ= E ( x )=(ap + bq ) 2
2
2
2
2
E ( X )=( a )( p )+( b )( q ) 2
E ( X )=( a p )+( b q ) E ( X ) =(a p + b q ) 2
2
2
x
¿ ¿
Var ( x )= E ( X X 2 )− E ¿ 2
2
2
Var ( X )=( a p + b q )−( ap + bq ) 2
σ =Var ( X )= pq ( a −b )
2
σ =√ pq pq ( a −b ) =|a −b|√ pq pq 2
5.72
En$! En$!en en" "e e /e# /e#
e0=n e0=n" "
σ x
2
σ x 2 7 e0+ e0+# #$# $#6n 6n
e 0#)!#e 0#)!#ene ne #0"#,! #0"#,!$#6n $#6n e e %0 (!n%0 (!n%0 %ne %ne
p + q =1 . x f(x)
μ 2 +"# "#n8
a
b q
)=(a )( p )+( b)( q ) E ( X )=( E ( x )=( )=( ap )+( bq ) μ= E ( x )=(ap + bq ) 2
2
2
2
2
E ( X )=( a )( p )+( b )( q ) 2
E ( X )=( a p )+( b q ) E ( X ) =(a p + b q ) 2
2
2
x
¿ ¿
Var ( x )= E ( X )− E ¿ 2
2
2
2
Var ( X )=( a p + b q )−( ap + bq ) 2
σ =Var ( X )= pq ( a −b )
2
σ =√ pq pq ( a −b ) =|a −b|√ pq pq 2
5.4' Se 0ee$$#%n 0ee$$#%n %0 $"0 e !n $ *!e $%n#ene 5 $"0 n!/e" n!/ e"0 0 2 2323 2323 7 ' Se X 0!/ 7 < e /=9#/% /=9#/% e %0 %0 n;/e"% n;/ e"%0 0 0ee$$ 0ee$$#%n #%n% %0. 0. En$!en En$!en"e "e #0" #0"#,! #,!$#6 $#6n2 n2 /e#2 /e#2 +"#n8 7 e0+#$#6n e0=n" e 0 +"#,e0 e%"#0 a) X F(x)
2 0.1
3 0.4
4 0.3
5 0.2
E ( x )=2 ( 0.1 )+ 3 ( 0.4 ) + 4 ( 0.3 ) + 5 ( 0.2 )
E ( x x )=3.6 E ( x )= 4 ( 0.4 ) + 9 ( 0.1 ) + 16 ( 0.2 ) + 25 ( 0.3 ) 2
2
E ( x )= 13.8 var ( x )= E ( x )− E ( x ) 2
2
var ( x )=13.8 −12.96 var ( x ) =0.84 σ =√ var var ( x ) σ =√ 0.84 = 0.91
b) *()
1 0.1
2 0.5
3 0.4
E ( y )= ( 0.1 ) + 2 ( 0.5 ) + 3 ( 0.4 ) E ( y ) =2.3 2 E ( y )=( 0.1 ) + 4 ( 0.5 ) + 9 ( 0.4 )
2
E ( y )=5.7
var ( y )= E ( y 2) − E ( y )
2
var ( y )=5.7 − 5.29 var ( y )= 0.41 σ =√ var ( y ) σ =√ 0.41 0.41=0.64
') +abla +abla ,e ,!"rbu' ,!"rbu' ju"a ,e X % x% 2 3 4 5
1 0.1 0 0 0
2 0 0.4 0.1 0
3 0 0 0.2 0.2
f(x) 0.1 0.4 0.3 0.2
*(%)
0.1
0.5
0.4
()
3 0.1
5 0.4
6 0.1
7 0.2
8 0.2
E ( z )=3 ( 0.1 )+ 5 ( 0.4 ) + 6 ( 0.1 ) + 7 ( 0.2 ) + 8 ( 0.2) E ( z )= 5.9 E ( z )= 9 ( 0.1 ) + 25 ( 0.4 ) + 36 ( 0.1 ) + 49 ( 0.2 )+ 8 ( 0.2 ) 2
2
E ( z )= 37.1 var ( z )= E ( z )− E ( z ) 2
2
var ( z ) =37.1− 34.81 var ( z )=2.29 σ =√ var var ( z ) σ =√ 2.29 2.29 =1.51
) ()
2 0.1
6 0.4
8 0.1
12 0.2
15 0.2
E ( w ) =2 ( 0.1 )+ 6 ( 0.4 ) + 8 ( 0.1 ) +12 ( 0.2 )+ 15 ( 0.2 ) E ( z )=8.8 E ( w )= 4 ( 0.1 )+ 36 ( 0.4 ) + 64 ( 0.1 ) + 144 ( 0.2 ) + 225 ( 0.2 ) 2
2
E ( w )= 95
var ( w )= E ( w )− E ( w ) 2
2
var ( w )=95 −77.44 var ( w )=17.56 σ = √ var var ( w ) 2.29 =4.19 σ =√ 2.29
DISTRIBUCIONES CONUNTAS2 VARIABLES ALEATORIAS INDEPENDIENTES 5.41.-C%n0#e"e #0"#,!$#6n $%n!n e X e < en ) 5.3' en$!en"e a)
E ( x ) y E ( y )
b)
Cov ( x , y )
')
σx ,σy y ρ ( X , Y ) σ
x% 1 5
-4 1/8 $ 3/8
2 1/4 1/8 5/8
7 1/8 1/8 1/4
1/2 1/2
a) E ( x )=( 1 )
() () 1
+ ( 5)
2
1 2
E ( x )=3 E ( y )= (− 4 )
() () () 3 8
+( 2 )
5
+ (7 )
8
1 4
E ( y )=1 b) E ( x , y )= (−4 ) (1 )
() 1 8
+( 2 ) ( 1)
() 1
4
+( 7 ) ( 1 )
() 1 8
(−4 ) ( 5 )
E ( x , y )=1.5
ov ( x , y )= E ( x , y ) – E ( x ) E ( y ) ov ( x , y )=1.5 – ( 3 )( 1 )
ov ( x , y )=1.5 ') E ( x 2 ) = ( 1 ) 2
() () 1 2
1
+( 5 ) 2
2
E ( x 2)= 13
var =13 – 9 var =4
σ =2 E ( y 2 ) =(−4 ) 2
() () () 3 8
+( 2 ) 2
E ( y 2 )=20.75
var =20.75−1 var =19.75
σ =4.4 ρ ( x , y )=1.5 /( 2 )( 4.4 )
ρ ( x , y )=0.17
5 8
+( 7 ) 2
1
4
() 1
4
+ ( 2) ( 5)
() 1 8
+ (7 ) ( 5 )
() 1 8
5.45.- C%n0#e"e #0"#,!$#6n $%n!n e X 7
Y en )!"
5.3'>,?. En$!en"e: X ) yE ( y ) (b) (a) E ( X x% 1 2 *(%)
-2 0.1 0.2 0.3
ov ( X , Y ) (') σ X σ Y % ρ ( X , Y ) . -1 0.2 0.1 0.3
4 0 0.1 0.1
5 0.3 0 0.3
F(x) 0.6 0.4
a)
∑ x F ( x )
E ( x )=
!
!
E ( x )=1 ( 0.6 ) + 2 ( 0.4 ) E ( x )=0.6 + 0.8 E ( x )=1.4 E ( y )=
∑ y F ( y ) !
!
E ( y )=−2 ( 0.3 )−1 ( 0.3 ) + 4 ( 0.1 )+ 5 ( 0.3 ) E ( y )=−0.6 −0.3 + 0.4 + 1.5 E ( y )= 1
b) ov ( X ,Y , Y ) ) = E ( X , Y ) )− E ( X ) ) E E ( Y )
E ( X X , Y ) =( 1 ) (−2 ) ( 0.1 ) + ( 1 ) (−1 ) ( 0.2 )+ ( 1 ) ( 5 ) ( 0.3 ) +( 2 ) (−2 ) ( 0.2 ) + ( 2 ) (−1 ) ( 0.1 )
+ ( 2 ) ( 4 ) ( 0.1 )=−0.2 −0.2 + 1.5−0.8 −0.2 + 0.8 =0.9 ov ( X ,Y , Y ) ) =0.9 − ( 1.4 ) ( 1 )=−0.5
(')
σ X
E ( x ) = 2
σ Y
∑ x
2
!
" ( x !)
E ( x ) =( 1 ) ( 0.6 ) + ( 2 ) ( 0.4 ) 2
2
E ( x ) =0.6 + 1.6 2
2
E ( x ) =2.2 2
var ( x )= E ( x x )−( E ( x )) 2
2
var ( x )=2.2 −(1.4 ) = 0.24 2
σ X =√ var var ( x )=√ 0.24 0.24 =0.489 E ( y )= 2
∑y
E ( y )=(−2 ) 2
2
!
2
( y !) " ( 2
2
2
( 0.3 )+ (−1 ) ( 0.3 ) + ( 4 ) ( 0.1 )+ ( 5 ) ( 0.3)
E ( y )=1.2 + 0.3 + 1.6 + 7.5=10.1 2
var ( x )= E ( y ) −( E ( y )) 2
2
var ( y )=10.1 −(1 ) =9.1 2
σ Y = √ var var ( y )=√ 9.1 9.1=3.01
ρ ( X , Y ) ) =
ov ( x , y ) −0.5 = =−0.3 σ X σ Y 3.01 ∗0.489
[email protected]!(%n) *!e 9 e 7 0%n +"#,e0 e%"#0 #ne(en#ene0 $%n 0 0#)!#ene0 #0"#,!$#%ne0 "e0(e$#+0: "e0(e$#+0: x F(x)
1 0.7
2 0.3
% F(%)
-2 0.3
#'ue"re la !"rbu' e x e % % erque que la x% 1 2
-2 0.21 0.09 0.3
5 0.35 0.15 0.5
8 0.14 0.06 0.2
0.7 0.3
5 0.5
ov ( x , y )=0 :
8 0.2
)=( 1 )( 0.7 )+( )+ ( 2)( 0.3) E ( x )=( E ( x )=1.3
)+ (8 )( 0.2 ) E ( y )=(−2 )( 0.3 )+( 5 )( 0.5)+( E ( y )=3.5 E ( x , y )=(1 )(−2 )( 0.21 )+( 1 )( 5 )( 0.35 )+( 1 )( 8 )( 0.14 )+( 2 )(− 2)( 0.09)( 2 )( 5 )( 0.15 )+( 2 )( 8 )( 0.06 )
E ( x , y )= 4.55 ov ( x , y )= 4.55 – ( 1.3)( 3.5 )
ov ( x , y )= 0
5.44.- C%n0#e"e #0"#,!$#6n $%n!n e X 7 < en )!" 531>?. >? En$!en"e E>X? 7 E>,? Dee"/#ne Dee"/#ne 0# X 7 < 0%n #ne(en#ene0 #ne(en#ene0 >$? En$!en"e $%+ >X2
2 0.06 0.14 3/8
3 0.15 0.35 5/8
4 0.09 0.21 1/4
f(x) 0.3 0.7
)=(1 )( 0.30 )+( 2 )( 0.70 ) E ( X )=( )= 0.30 + 1.4 E ( X )= )= 1.7 E ( X )= E ( Y )=(2)( 3 / 8 )+( 3 )( 5 / 8 )+( 4 )( 1 / 4 ) E ( Y )=3 / 4 + 15 / 8 + 1
E ( Y )=29 / 8=3.1
(b) : ! eee"e!
( )
,ebe !er !'re"! % x e % ! eee"e! E ( X , Y )=(1)( 2 )( 0.06 )+( 1 )( 3 )( 0.15 )+( 1 )( 4 )( 0.09 )+( 2 )( 2)( 0.14 )+( 2 )( 3 )( 0.35 )+( 2 )( 4 )( 0.21 )
E ( X , Y )=0.12 + 0.45 + 0.36 +0.56 + 2.1 + 1.68 E ( X , Y )=5.27
Cov ( X ,Y )= E ( X , Y )− E ( X ) E ( Y ) Cov ( X , Y )=5.27 −( 1.7 )( 3.1)
Cov ( X ,Y )= 0 P# P # 1−
1 2
2
P# P # 0.75
5.4 5.4..-C% C%n0 n0# #e" e"e e #0 #0"# "#,! ,!$# $#6n 6n $%n $%n!n !n e 9 e 7 en ) )!" !" en$!en"e: a)
E ( x ) yE ( y ) $
,? Dee"/#n Dee"/#ne e 9 e 7 0%n #ne(en #ne(en#en #ene0. e0. $? En$!en"e En$!en"e #0"#,! #0"#,!$#6n $#6n /e# /e# 7 e0+#$#6n e0+#$#6n e0=n" e0=n" e +"#,e. x% 0 1 2
-2 0.05 0.1 0.03
-1 0.05 0.05 0.12
0 0.1 0.05 0.07
1 0 0.1 0.06
a)
)=( 0)( 0.30)+( )+ ( 1)( 0.35)+( )+ ( 2 )( 0.35 ) E ( x )=( E ( x )=1.05 E ( y )=(−2 ) ( 0.18 )+ ( −1 ) ( 0.22 ) + ( 0 ) ( 0.22 )+ ( 1 ) ( 0.16 )
+(2 )( 0.08 )+( 3 )( 0.14 ) E ( y )=0.16
2 0.05 0 0.03
3 0.05 0.05 0.04
b) (0.3)(0.18) = 0.05 0.054 = 0.05
! eee"e! ee e"e!
(0.30)(0.22) = 0.05 0.066 = 0.05 ')
-2 0. 0.05
-1 0.15
0 0.18
1 0.17
2 0.22
3 0.11
4 0.08
E ( z )= (−2 ) ( 0.05 ) + (− 1 ) ( 0.15 ) + ¿
(0 )( 0.18 )+( 1 )( 0.17 )+( 2)( 0.22 )+( 3 )( 0.11 ) +( 4 )( 0.08 )+( 5 )( 0.04 ) E ( z )=1.2
)+ ( 0 ) 2 ( 0.18)+( )+ ( 1) 2 ( 0.17 ) E ( z 2)=(−2) 2 ( 0.05 )+(−1 ) 2 ( 0.15 )+( +(2 ) 2 (0.22)+( )+ ( 3 ) 2 ( 0.11 )+( 4 )2 ( 0.08)+( )+ ( 5 )2 (0.04 ) E ( z 2)= 4.67
Var ( z )= 4.37 – ( 1.21 ) 2 Var ( z )= 3.21 3.21 σ =√ 3.21
σ =1.79
5.4.- Un /%ne e*!##," 0e n8 1 +e$e0 0e X e n!/e"% e $"0 $"0 *!e %$!""en %$!""en 7 0e < 0e$!e 0e$!en$# n$# e $"0 $"0 /0 /0 ") ") *!e *!e %$!""e. Dee" Dee"/#n /#ne e !n$#6 !n$#6n n $%n $%n!n !n e e X 7 < , En$! En$!en en" "e e $%+> $%+>X2 X2 (>X2 X2
0 1/16 0 0
1 0 4/16 3/16
2 0 0 3/16
3 0 0 0
4 0 0 0
f(x) 1/6 4/16 6/16
5 0.04
3 4 *(%)
0 0 1/16
0 0 7/16
2/16 0 5/16
2/16 0 2/16
0 1/16 1/16
4/16 1/16
b)
( )
E ( x , x )=
1
16
+
4 16
+
6 16
+ 12 + 12 + 18 +1=5.41 16
16
16
ov ( x , x )= E ( x , x )− E ( x ) E ( y ) ov ( x , x )=5.41 −( 2∗1.7 ) = 0.85
ρ ( x , x )=
ρ ( x , x )=
ov ( x , y ) σ x σ y 0.85
( 0.64 ) ( 1.7 )
ρ ( x , x )=0.89
5..-Se 0ee$$#%nn %0 $"0 8" e !n $ *!e $%n#ene $#n$% $"0 n!/e"0 2 2 32 3 7 ' 0e 9 0!/ 7 7 /=9#/% e %0 3 n;/e"%0 0$%0 ? Dee"/#n Dee"/#n" " #0"#,! #0"#,!$#6n $#6n $%n! $%n!n n e 9 e 7 ,? En$!en"e En$!en"e $%+>92 $%+>927? 7? 7 >927? a) x% 2 3 4 5
1 0.1 0 0 0
2 0 0.4 0.1 0
3 0 0 0.2 0.2
E ( x )=( )=(2 )( 0.1 )+( 3 )( 0.4 )+( 4 )( 0.3 )+( 5 )( 0.2 ) E ( x )=3.6
)+ ( 3)( 0.4 ) E ( y )=(1)( 0.1)+( 2)( 0.5 )+( E ( y )=2.3
)+ ( 3 )( 4 )( 0.2)+( )+ ( 3)( 5)( 0.2)+( 2)( 4 )( 0.1 ) E ( x , y )=(1 )( 2)( 0.1 )+( 2)( 3 )( 0.4 )+(
E ( x , y )= 8.8
Cov ( x , y )= E ( x , y ) – E ( x ) E ( y ) Cov ( x , y )= 8.8 – ( 3.6)( 2.3)
Cov ( x , y )= 0.52
)+ ( 4 ) 2 ( 0.3 )+( )+ ( 5) 2 ( 0.2 ) E ( x 2)=(2 ) 2 ( 0.1)+( 3) 2 ( 0.4 )+( E ( x 2)= 13.8 σ 2=13.8 – ( 3.6 ) 2
σ 2=0.84 σ =√ 0.84 = 0.92
)=( 1 ) 2 (0.1)+( )+ ( 2) 2 ( 0.5 )+( 3) 2 ( 0.4 ) E ( y 2 )=( E ( y 2 )=5.7
σ 2=5.7 – ( 2.3 ) 2 σ 2=0.41 σ =√ 0.41 =0.64 ρ ( x , y )= ov ( x , y )/ σxσy
ρ ( x , y )=0.52 /( 0.92 )( 0.64 ) ρ ( x , y )=0.9
DESIGUALDAD DE CHEB
σ . U##$e e0#)! e Ce,70e+ (" e0#/"
P ( μ−3 σ % μ +3 σ ) . Pr el "ere;a< P ( μ μ − kσ % X % μ + kσ ) # 1 −
1 2
k
P ( μ μ −3 σ % X% μ + 3 σ ) # 1−
1
P ( μ μ −3 σ % X% μ + 3 σ ) # 1−
1
2
3
9
P ( μ μ − 3 σ % X % μ + 3 σ ) ) # 0.888
5.3.-Sen 8 +"#,e e%"# n%"/ e0=n" $%n /e#
u= 0
σ =1 . U##$e e0#)! e $e,70e+
7 e0+#$#6n e0=n"
(" en$%n"" !n +%" , (" e $! P (−b≤ z ≤b )≥0.9 1 – 1
/ k 2=0.9
0.1=1 / k 2
& = √ 10 10
b = kσ b = √ 10 10 ( 1) 10 b = √ 10
5.'.5.'.- Se Se e0=n e0=n" "
!n !n +"#, +"#,e e e% e%"# "# $%n $%n /e# /e# σ x =1 $ 5
. U# U##$ #$e e e0# e0#)! )! e Ce, Ce,70 70e e+ + (" ("
e0#/": P >-' % X ¿ '?. P ( μ −k σ % X % μ + k σ ) # 1−
1 2
k
P ( 0− 1.5 k % X % 0 + 1.5 k ) # 1−
0
−1.5 k =− =−3
−1.5 k =−3
μ=0 7 e0+#$#6n
1 2
k
k = 2
P# P # 1−
+
1 2
k
=3
0 1.5 k
1.5 k = 3
k =2
5.1.-Se 9 !n +"#,e e%"# $%n /e#
u=70 J(" *!e +%"
e σ ("%!e" e0#)! e $e,70e+ P ( 65 ≤ X ≤75 )≥ 0 $ 95 K 1
−1 / & = 0.95 =1 / &
0.05
& = √ 20 20
u− kσ = 65
− √ 20 20 σ =65
70
σ =
5 20 √ 20
=1.12
5.5.- Se X !n +"#,e +"#,e e%"# e%"# $%n /e# e0=n"
σ =10 . U# U###$e e0#) e0#)! ! e Ce,7 ,70 0e+ (" ("
e0#/":
P ( X # 120 )
,
P ( X % 75 )
,a"! u=100
σ =10 a)
u=100 7 e0+#$#6n e0+#$#6n
P (u − kσ ) % X % (u + kσ ) # 1 −
1 2
k
u− kσ =120
− k 10 =120
100
k 10=100 −120
k =
100
−120
10
k =−2 1−
1
(−2 )2
= 0.75
P ( X # 120 ) =075
b)
P (u −kσ ) % X % (u + kσ ) # 1 −
1 2
k
u + kσ =75 100
+ k 10= 75
k 10= 75 −100
k =
75
−100 10
k =−2.5 1−
1
(−2.5 )2
= 0.84
P ( X % 75 ) = 0.84
PROBLEMAS MISCELNEOS
5.@ 5.@..-Se Sen n 9 !n !n +"# +"#, ,e e e e% %"# "# $%n $%n#n #n! ! $%n $%n 0#)! 0#)!#e #en ne e #0"#,!$#6n
/
1 8 '! 0
( x ) " (
≤ x ≤8
0 ()o*ra'
par*('
En$!en"e: a) P ( 2≤ x ≤5 ) ,? P>'Q P>'Q9 9Q Q4? 4? $? P>9@? a)
+ = b∗
() 1
+ = (3 )
+ =
8
3 8
b) + = ( 4 )
() 1 8
+ = ½
') + = (2 )
() 1 8
+ = ¼
5.4.- Dee"/#ne 7 "$e )"$ e !n$#6n e #0"#,!$#6n $!/! F e +"#,e e%"# X e ("%,e/ 5.@.
" ( ( x )=
{
1 8
'! 0 % x % 8
0 ()o*ra'par*('
P ( 2 % X % 5 ) (b) P ( 3 % X % 7 ) (') P ( X # 6 ) .
a
{
0 x < 0 1 16
X 0 % x % 8
1 x > 8
Pr '!gue"e '!gu e"e b"ee;! b"ee;! ua fu' e rbabla a'u;ula"a<
F ( x ) ('!-ua.a : 0 '!x < 0,
X 8
'! 0 % x % 8, y 1 '!x > 8
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>9?
0 ()o*ra
E+! 7 en$!en"e a) P ( 1≤ x ≤3 ) b)
P ( 2≤ x ≤ 4 )
')
P ( x ≤3 )
+ = b∗ / 2 1=( 5 k )( 5) / 2
& =2 / 25
par*(
a)
( 1)= 2 / 25 " ( ( 3)= 6 / 25 " ( a =½
a=
(
2 25
+
6 25
)( ) 2
8 25
b)
( 2)= 4 / 25 " ( " ( ( 4 )= 8 / 25
a =½
(
4 25
+
8 25
)( ) 2
a = 12 / 25
')
( 3)= 6 / 25 " ( ( 0 )=0 " (
(= ) ( ) = 6
a
25
2
3
9 25
5..- G"*!e !n$#6n e #0"#,!$#6n $!/! F e +"#,e e%"# #0$"e X $%n 0#)!#ene #0"#,!$#6n: x f(x)
-3 1/4
2 1/2
6 1/4
1
1
1
4
2
4
F ( X ) = ∗ μ−1 ( x + 3 ) + ∗ μ−1 ( x −2 ) + μ− 1 ( x − 6 )
5..-P"!e,e e e%"e/ 5. 0e x , y , z +"#,e0 e%"#0 e S $%n / =ф ( x , y ) en%n$e0 ❑
E>Z?
∑ ф ( x! , y0 ) ( x! , y0 )
%ne e0 #0"#,!$#6n $%n!n e 9 e
!, 0
7 X = X! , 1 1 $ X) Y =Y!11 .. Y2
/ =ф ( x , y ) ❑
∑ ( x! , y0 )
)= - ( / )=
!,0
❑
❑
❑
/ ∑ ( X! , Y0 ) ∑ /- ( /0) =∑ ❑ ❑
)= E ( / )=
!,0
❑
( X! , Y0 )
❑
/ =¿ ∑ ф ( x , y ) ( X! , Y0 ) ∑ ❑ ❑ ❑
∑¿
E ( / )=
!,0
5..- Se X !n +"#,e +"#,e e%"# e%"# (" e $! *!e (>929? 7 (>92-9?- x f(x)
1 0.1
2 0.5
3 0.4 E ( x )=2.3
σ =√ 0.41 0.41=0.64
xx 1 2 3 F(x)
1 0.1 0 0 0.1
2 0 0.5 0 0.5
3 0 0 0.4 0.4
f(x) 0.1 0.5 0.4
-3 0 0 0.4 0.4
f(x) 0.1 0.5 0.4
E ( x , x )=( 0.1 ) + 2 + 3.6 =5.7 ov ( x , x )= E ( x , x )− E ( x ) E ( x ) ov ( x , x )=5.7 −( 2.3∗2.3 )=0.41
ρ ( x , x )=
ρ ( x , x )=
ov ( x , y ) σ x σ x
0.41
( 0.64 ) ( 0.64 )
ρ ( x , x )=1
xx 1 2 3 F(x)
-1 0.1 0 0 0.1
-2 0 0.5 0 0.5
σ x 3 0
e/!e0"e
E ( x , x )=−( 0.1 )−2− 3.6=−5.7
ov ( x , x )= E ( x , x )− E ( x ) E E ( x ) ov ( x , x )=−5.7 + ( 2.3∗2.3 )=−0.41
ρ ( x , x )=
ov ( x , y ) σ x σ x
ρ ( x , x )=
−0.41 ( 0.64 ) ( 0.64 )
ρ ( x , x )=−1
5.3.- Se X !n +"#,e e%"# (" e $! *!e (>929? 7 (>92-9?- x f(x)
1 0.1
2 0.5
3 0.4 E ( x )=2.3
0.41=0.64 σ =√ 0.41
xx 1 2 3 F(x)
1 0.1 0 0 0.1
2 0 0.5 0 0.5
E ( x , x )=( 0.1 ) + 2 + 3.6 =5.7 ov ( x , x )= E ( x , x )− E ( x ) E ( x ) ov ( x , x )=5.7 −( 2.3∗2.3 )=0.41 ρ ( x , x )=
ov ( x , y ) σ x σ x
3 0 0 0.4 0.4
f(x) 0.1 0.5 0.4
σ x 3 0
e/!e0"e
ρ ( x , x )=
0.41
( 0.64 ) ( 0.64 )
ρ ( x , x )=1
xx 1 2 3 F(x)
-1 0.1 0 0 0.1
-2 0 0.5 0 0.5
E ( x , x )=−( 0.1 )−2− 3.6=−5.7 ov ( x , x )= E ( x , x )− E ( x ) E ( x ) ov ( x , x )=−5.7 + ( 2.3∗2.3 )=−0.41 ρ ( x , x )=
ov ( x , y ) σ x σ x
ρ ( x , x )=
−0.41 ( 0.64 ) ( 0.64 )
ρ ( x , x )=−1
-3 0 0 0.4 0.4
f(x) 0.1 0.5 0.4
