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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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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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)

= ( =