Ejercicio 3.3 y 3.6
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Transcript of Ejercicio 3.3 y 3.6
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7/29/2019 Ejercicio 3.3 y 3.6
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Estadstica y Probabilidad
Ejercicio 3.31.
a)Variable aleatoria discreta [0,100]b)Variable aleatoria continua *0,+c) Variable aleatoria discreta *1,2,3+d)Variable aleatoria discreta [0, +e) Variable aleatoria continua [0, +
f) Variable aleatoria continua [0, +g)Variable aleatoria discreta *1,2,3+h)Variable aleatoria continua [0, ]3.
= {1,2,3,4,5,6}
Pr(A) =
Pr(B) =
0
0.5
1
1.5
0 1 2
F(x)
X 0 1
Pr(X=xi)
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X 0 1
Pr(X=x)
5. X= [# de carros defectuosos entre los escogidos]
k={0,1,2}
Pr(x=0)= =
= Pr(x=1)=
=
=
Pr(x=2)= =
= X 0 1 2
Pr(X=k)
0
0.5
1
1.5
0 1 2
F(x)
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7.
a) Si Pr(x) =
x 1 2 3 4
Pr (x = xi)
=
b)
x 1 2 3 4
Pr (x = xi)
=
- =
9.
a) Para que sea una funcin de probabilidad la sumatoria de probabilidades debe ser igual a 1
0.301 + 0.1761+0.125+0.097+0.079+0.067+0.058+0.051+0.0457 = 1
Por lo tanto si es una funcin de probabilidad
b)W(1,3,5,7,9) Pr(x== 0.301+0.125+0.079+0.058+0.0457 = 0.6089
1 2 3 4 5 6 7 8 9Pr(x=k) 0.301 0.1761 0.125 0.097 0.079 0.067 0.058 0.051 0.0457
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c)
11.
a)
b)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10
)1,0(
......2,1,0
1)Pr(
p
k
ppkXk
11
11
*
111
1*
111
1*
11
1)1(
0
0
pp
pp
pp
pp
pp
k
k
k
k
21)1()1(
10)1(
0
)(01
0
tsipppp
tsipp
otsi
tF
0
0)1(
00
)(
k
k ksipp
ksi
kF
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c)
13.
a) Pr(1x5) = Pr(x5) - Pr(x1)= F(5) F(1)
= b) Pr(2x4) = Pr(x4) - Pr(x2)
= F(4) F(2)
= c) Pr(0x3) = Pr(x3) - Pr(x0)
= F(3) F(0)
=
d) Pr(4x6) = Pr(x6) - Pr(x4)= F(6) F(1)
= 15.
a) F(x) =
Si x/6
F(x) = = 0
3
32
322
22
210
1)2Pr(
331)2Pr(
21)2Pr(
)21(1)2Pr(
)1()1()1(1)2Pr(
)2Pr(1)2Pr(
pX
pppX
ppppppX
ppppppX
ppppppX
XX
4
432
32322
3222
3210
1)4Pr(
4641)4Pr(
3321)4Pr(
)331()21(1)4Pr(
)1()1()1()1(1)4Pr(
)4Pr(1)4Pr(
pX
ppppX
ppppppppppX
ppppppppppX
ppppppppX
XX
)33()3Pr(
33)3Pr(
2)3Pr(
)21()3Pr(
)1()1()1()3Pr(
)3Pr()3Pr(
2
32
322
22
210
pppX
pppX
ppppppX
ppppppX
ppppppX
XX
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Si /6 < x /3
F(x) = = -cos3x
Si x>/3
F(x) = F(x) = = 1
0 si x/6
F(x)= -cos3x si /6 < x /3
1 si x>/3
b) Pr(x=0.2) = 0Pr(x /4) = F( /4) = -cos(3/4 ) = /2Pr(x> /3) = 1- F( /3) = 1 + cos() = 0
Pr(/12x ) = F() - F(/12) = -cos (3 ) + cos(/4) =
17.
a) = =
b + 1 = 3
b=2
1= 1= a=1
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b)3z2 si
f(z)=
0 caso contrario
0 si F(z)= z3 si 1 si
19.
a)
b)
40
416
1
102
1
)(
t
t
t
tf
2
141Pr
6
3)41Pr/(
6
1
6
4)41Pr(
6
1)41Pr(
6
1)41Pr(
).()41Pr()1Pr(
4
1
4
1
4
1
t
t
t
tt
dtt
dttftt
3
1)2Pr(
6
2)2Pr(
2
1
6
1
1)2Pr(
2
1
6
11)2Pr(
.2
1
6
11)2Pr(
).().(1)2Pr(1)2Pr(
1
0
2
1
1
0
2
1
2
1
1
0
t
t
t
ttt
dtdtt
dttfdttftt
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c)
21.
,
Pr(x=1) = (1/25)/ = 1/25
Pr(x=2) = (4/25-1/25)/ = 3/25
Pr(x=3) = (9/25-4/25)/ = 5/25
Pr(x=4) = (16/25 -9/25)/ =7/25
Pr(x=5) = ( -16/25)/ =9/25
a) Puntaje del jugador
b) La ganancia del jugador
xi 0 1 2 3 4 5
Pr(x=xi) 1/25 3/25 5/25 7/25 9/25 0
Gi 5 4 3 2 1 0
Pr(x=Gi) 1/25 3/25 5/25 7/25 9/25 0
6
1)3Pr(
3
1
6
11)3Pr(
6
1
2
11)3Pr(
6
1.
2
11)3Pr(
).().(1)3Pr(1)3Pr(
3
1
1
0
3
1
1
0
3
1
1
0
t
t
ttt
dtdtt
dttfdttftt
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23. c x si 0x
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c) No son independientesPr(y=3x=1) Pr(y=3)*Pr(x=1)
d)
Pr(y=1)= Pr(y=2)= Pr(y=1)=
27.
{ [] []T=-3z
F(x)= F(x)=
{
[]
{
[]
Entonces:
{ []
y 1 2 3
Pr(x=xi) 2/18 5/18 11/18
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{ []
29.
1/3 si [ -1, 2]
Sea: 0 caso contrario
Hallar si Si entonces por lo tanto:
1/3 1/6 1/3
0 caso contrario
31.
Fx(x)=1-e-x
si x0
a) Y=Pr(x=Fx(t))= [1-e-t si t0 +
Fy(t)=Pr(yt) = Pr(t)= Pr(xt2)
= Fx(t2)-Pr(x= t2)
f= F x
(1-e-t
)= e-t
(1-e-t )=-2t
Fx(t)= Pr(xt)
Fx= Pr(t)Fy(t)= Pr(xt2)
Fy(t)=Fx(t2)= 1-e-t2
fy(t) = 2 t e-t2
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b) Z= Fx(t)= 1-e-t
Z=lnx Z=lnxX=e
z
lnxt
1/lnx t
x et
Fx(t)=Pr(xt)
= Pr(ez t)
=Pr(xet) =1-e^-tt
f= F x
(1-e^-tt)=2 et e-te^
(1-e^-tt)= 2 et-te^
fy(t) = 2 et-te^
Ejercicio 3.61.
X -0.71 0.24 0.61
p 0.2 0.5 0.3
E(x) = E(x) = 0.161
E(x2) = (-0.71)
2(0.2)+(0.24)
2(0.2)+(0.61)
2(0.3)
E(x2) = 0.24125
Var (x) = E(x2) - (E(x))2
Var (x) = 0.24125 (0.161)2 = 0.2154
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Y 2 4 5 6
p 0.3 0.1 0.2 0.4
E(y) = 2(0.3)+4(0.1)+5(0.2)+6(0.4)
E(y) = 4.4
E(y2) = (2)2(0.3)+(4)2(0.1)+(5)2(0.2)+(6)2(0.4)
E(y2) = 22.2
Var (y) = E(y2) - (E(y))
2
Var (y) = 22.2 (4.4)2
= 2.84
3.
E (X-2) = 8
E(x) E(2) = 8
E(x) = 10
E ((X+1)2) = 120
E(x2 + 2x + 1) = 120
E(x2)+E( 2x) +E(1) =120
E(x2) = 120- 2E(x)-1
E(x2) = 120 2(10) -1
E(x2) = 99
Var(x) debe ser positiva
Var (x) = E(x2)- (E(X))
2
Var (x) = 99 (10)2
Var (x) = -1
No existe una variable X que cumpla.
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7/29/2019 Ejercicio 3.3 y 3.6
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5.
a) S = x + y + zE (5) = E (x) + E (y) + E (z)
= u + u + u=3 u
V (5) = V(x + y + z)
= V(x) + V(y) + V(z)
= b) T = 3x
E (x) 3 u
Var(x) = Var (x)= 9
c) A = E (A) =
= uVar (A) = Var }
() ( ) =
d) Var ( 5) = E - (E ( E () = Var (5) + (E ( + ( Var ( A) = E ( E (
=
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7.
a)f(x) =
E(x) = E(x2) = V(x) = E(x2) E(x)2 = 7
b)f(x) =
E(x) = E(x
2) =
V(x) = E(x2) E(x)2 =
c) E(x) = E(x
2) =
V(x) = E(x2) E(x)2 =
si 0 caso contrario
si 0 caso contrario
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d) E(x) =
E(x2) =
V(x) = E(x2) E(x)
2=
e) F(x) = 1 = E(x) =
E(x2) =
V(x) = E(x2) E(x)
2=
9.
z a 2ap 0.5 0.5
E(z) = a(0.5)+2a(0.5) = 0.9
3a = 1.8
a = 0.6
E(z2) = (0.6)
2(0.5)+(0.12)
2(0.5) = 0.9
Var (z) = E(z2) - (E(z))
2
Var (z) = 0.9 (0.9)2
= 0.09
11.
X -1 0 1
p P1 P2 P3
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E(x) = -1 P1 + 0 P2 + 1 P3 = 0.1 E(x2) = P1 + 0 P2 + 1 P3 = 0.8
P3 - P1 = 0.1 P3 + P1 = 0.8
P1 = 0.35
P3 = 0.45
PT = P1 + P2 + P3
1 = 0.35 + P2 + 0.45
P2 = 0.2
13.
F (x) = {
para
a) Pr = 0Pr F (4) F (2)
4a +6 2a b =
2a = ax + b
2a = 4a + b
2a = b
a =
Entonces a = 0.5
b = 1
b) F (x) { 2 c) Pr
= 0.5 0 = 0.5
Pr ( = 1 Pr = 1 F (2.5) + Pr ( x = 2.5)
= 1 - (0.5 + 0.25)
= 0.25
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d) E (x) = Var (x) = =
15.
a) Para X = 0 Para X = 1
Para X = 2x 0 1 2
Pr
b) E (X) =
c) E (x
2) =
Var (X) = E (X2) (E (X))
2
Var (X) = - ( )2
Var (X) =
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17.
a) = * (A); (F,A); (F,F,A); (F,F,F) }b)
+
*
+
*
*
=
c)G 10000 1000 -500 0
p
E(G) = * (10000) + * (1000) + * (-500)
E(G) = 2133.34
d)
x 1 2 3
p
E(x) =
+
* (2) +
* (3)
E(x) = Var(x) = E(x
2) (E(x))
2
E(x2) =
+ (2)2 * + (3)2 * = Var(x) =
- ()2Var(x) = 0.62
19.
a)
x 0 3000 5000
Pr 0.1 0.4 0.5
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b) E( c) = E( c) = (0.1)(0) + (0.1)(3000) + (0.5)(5000)
E( c) = 3700
21.
Compaa Ganancia Probabilidad
A -200 0,0102
B -150 0,0346
C -50 0,0860
D 50 0,1542
E 150 0,2123
F 250 0,2123
G 350 0,1542
H 450 0,0860
I 550 0,0346
J 650 0,0156
a) EsperanzaE(x)= pk*xk= 202,94
b) VarianzaVar(x)= E(x2) (E(x))2= 32009,8564
c) Desviacin Estndar = = 178,91298623 .
a) x2, Si 0 x 4
f (x) =
0, caso contrario
E(X) = ( X2) dxE(X) =
dx
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E(X) = (
)
E(X) = ( )
E(X) = 3
(X2) = ( X2) dx Var (X) = E (X2) (E (X))2
E(X2) =
dx Var (X) = - (3)2E(X2) =
(
) Var (X) = E(X
2) =
( )E(X
2) =
= = = 0,7746
b) Si Y = 2X 0,5
E (Y) = E (2X 0,5)
E (Y) = 2E(X) 0,5
E (Y) = 2 (3) 0,5
E (Y) = 5,5
E (Y2) = E (2X 0,5)2
E (Y2) = E (4X
2 2X + )
E (Y2) = 4E(X2) 2E(X) +
E (Y2) = 4 ( ) 2 (3) +
E (Y2) =
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Var (Y) = E (Y2) (E (Y))
2
Var (Y) = - (5,5)2
Var (Y) =
25.
contrariocaso
xsixx
xF
0
801024
83
)(
2
a)
hxE
xE
xxxE
xxxE
xxxE
dxxx
xxE
dxxfxxE
8.4)(
5
88*2
1024
3)(
54
8
1024
3)(
81024
3)(
)8(1024
3
)(
.1024
)8(3.)(
).(.)(
54
8
0
54
8
0
43
8
0
3
8
0
2
8
0
2
2
22
2
652
8
0
652
8
0
542
8
0
42
8
0
222
8
0
22
56.2)(
8.46.25)(
)()()(
6.25)(
6
8
5
8*8
1024
3
)(
65
8
1024
3)(
81024
3)(
)8(1024
3)(
.1024
)8(3.)(
).(.)(
hxVar
xVar
xExExVar
xE
xE
xxxE
xxxE
xxxE
dxxx
xxE
dxxfxxE
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b)
27.
E(x)=PkXk
Pr(x=k)=pqk sea q=1-p
R= = =pE(x)=PkXk=
=V(x)=E(x
2)-(E(x))
2
V(x)=- =
29.
X= (3,1,2)
Pr(x=3) = t -
Pr(x=1) = t -
Pr(x=2) = t
t - + t - t = 1
3t =
t=
E(x) =
48)(
2*5*8.4)(
cE
cE
42.14
)(
208)(
48256)(
256)(
2*5*6.25)(
2
2
cVar
cVar
cVar
cE
cE
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7/29/2019 Ejercicio 3.3 y 3.6
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Y = (2,1,0)
Pr(x=2) = a
Pr(x=1) = b
Pr(x=0) = c
2b + b + 2c = b 2b = a b =
c= a=
E(y) = 2( =
E(XY + 2Y X) = E(X)E(Y) + 2E(Y)E(X)
= ( =