STAT 400 Joint Probability Distributions

STAT 400

Joint Probability Distributions

Fall 2017

1. Let X and Y have the joint p.d.f. f X , Y ( x, y ) = C x 2 y 3, 0 < x < 1, 0 < y < x , zero elsewhere.

a) What must the value of C be so that f X , Y ( x, y ) is a valid joint p.d.f.?

b) Find P ( X + Y < 1 ).

c) Let 0 < a < 1. Find P ( Y < a X ).

d) Let a > 1. Find P ( Y < a X ).

e) Let 0 < a < 1. Find P ( X Y < a ).

f) Find f X ( x ). h) Find f Y ( y ). j) Find E ( X Y ).

g) Find E ( X ). i) Find E ( Y ). k) Find Cov ( X, Y ).

l) Are X and Y independent?

2. Let X and Y have the joint probability density function

f X , Y ( x, y ) =

1,

x

x > 1, 0 < y < 1 , x

zero elsewhere.

a) Find f X ( x ).

b) Find E ( X ).

c) Find f Y ( y ).

d) Find E ( Y ).

3. Let X and Y have the joint probability density function

f X , Y ( x, y ) =

1,

x

0 < x < 1, 0 < y < x,

zero elsewhere.

a) Find f X ( x ).

b) Find E ( X ).

c) Find f Y ( y ).

d) Find E ( Y ).

e) Find P ( X + Y 1 ).

f) Find Cov ( X, Y ).

4. Let X and Y be two random variables with joint p.d.f.

f ( x, y ) = 64 x exp{ ? 4 y } = 64 x e4 y ,

0 < x < y < ,

zero elsewhere.

a) Find P ( X 2 > Y ).

b) Find the marginal p.d.f. f X ( x ) of X. c) Find the marginal p.d.f. f Y ( y ) of Y. d) Are X and Y independent? If not, find Cov ( X, Y ) and = Corr ( X, Y ).

e) Let a > 1. Find P ( Y > a X ).

f) Let a > 0. Find P ( X + Y < a ).

5. Let the joint probability mass function of X and Y be defined by

p ( x, y ) =

x y ,

32

x = 1, 2,

y = 1, 2, 3, 4.

a) Find P ( Y > X ).

b) Find p X ( x ), the marginal p.m.f. of X. c) Find p Y ( y ), the marginal p.m.f. of Y. d) Are X and Y independent? If not, find Cov ( X, Y ).

6.

Let the joint probability mass function of X and Y be defined by

p ( x, y ) =

x y

30

,

x = 1, 2,

y = 1, 2, 3, 4.

a) Find P ( Y > X ).

b) Find p X ( x ), the marginal p.m.f. of X. c) Find p Y ( y ), the marginal p.m.f. of Y. d) Are X and Y independent? If not, find Cov ( X, Y ).

7. Suppose the joint probability density function of ( X , Y ) is

f

x,

y

C

x

0

y

2

0 y x1

otherwise

a) Find the value of C that would make f x, y a valid probability density function.

b) Find the marginal probability density function of X, f X ( x ).

c) Find the marginal probability density function of Y, f Y ( y ).

d) Find P ( X > 2 Y ).

e) Find P ( X + Y < 1 ).

f) Are X and Y independent? If not, find Cov ( X, Y ).

8. Let X and Y have the joint probability density function

f ( x, y ) = C x, 0 x 1, 0 y x ( 1 ? x ),

a) Find the value of C so that f ( x, y ) is a valid joint p.d.f.

b) Find f X ( x ).

c) Find E ( X ).

d) Find f Y ( y ).

f) Are X and Y independent?

e) Find E ( Y ).

zero elsewhere.

9. Let X and Y have the joint probability density function

f X, Y ( x, y )

=

x4y

0

0 y x 1

otherwise

a) Find P ( X > 4 Y ).

b) Find the marginal probability density function of X, f X ( x ). c) Find the marginal probability density function of Y, f Y ( y ). d) Are X and Y independent? If not, find Cov ( X, Y ).

e) Find P ( Y > 0.4 | X < 0.8 ).

f) Find P ( X < 0.8 | Y > 0.4 ).

1. Let X and Y have the joint p.d.f. f X , Y ( x, y ) = C x 2 y 3, 0 < x < 1, 0 < y < x , zero elsewhere.

a) What must the value of C be so that f X , Y ( x, y ) is a valid joint p.d.f.?

1

0

x

C

0

x2

y3

dy

dx

=

1

0

C 4

x4

dx

=

C

20

= 1.

C = 20.

b) Find P ( X + Y < 1 ).

y = x and y = 1 ? x

x = y 2 and x = 1 ? y

y =

5 2

1

.

P(X + Y < 1) =

5 1

2

1 y

20

x2

y3

dx

dy

0 y 2

5 1

=

2

0

20 3

1

y 3

y3

20 3

y9

dy

5 1

=

2

0

20 3

y 3

20 y 4

20 y 5

20 3

y6

20 3

y9

dy

=

5 3

y 4

4

y5

10 3

y6

20 21

y7

2 3

y 10

5 1

2 0.030022.

0

OR

y < x and y = 1 ? x

x =

5 2

1

2

1

5 1 2

3 2

5

.

P(X + Y < 1)

=

1

1

3

5

x

20

1 x

x

2

y3 dy

dx

2

1

= 1 5 x 4 5 x 2 1 x 4 dx

3 5

2

1

= 1 5 x 2 20 x 3 25 x 4 20 x 5 5 x 6 dy

3 5

2

=

1

5 3

x

3

5

x

4

5

x

5

10 3

x

6

5 7

x

7

1 3 5

0.030022.

2

c) Let 0 < a < 1. Find P ( Y < a X ).

P(Y < aX)

=

1

0

a x

20

0

x2

y3

dy

dx

=

1

5 a 4 x 6 dx

0

=

5 7

a

4

.

d) Let a > 1. Find P ( Y < a X ).

y = x and y = a x

x =

1

a2

,

y= 1. a

P(Y < aX) =

1a

1

ya

20

x 2

y3

dx

dy

0

y2

=

1a

1 0

20 y 6 3a3

20 3

y9

dy

=

1

2

7 a 10

.

P( Y < a X )

=

1

1

a2

x

20

x 2

y3

dy

dx

1 a2

= 1

0 a x

0

5x4 5a 4 x6

dx

=

1

7

2

a 10

.

e) Let 0 < a < 1. Find P ( X Y < a ).

y = x and y = a x

x = a 2 3.

P(XY < a)

=

1

1 a2 3

x

20

ax

x 2

y3

dy

dx

=

1

1

a2 3

5

x

4

5

a x

4 2

dx

=

1

x5

5

a4 x

1

a2 3

= 6 a 10 3 5 a 4 .

f) Find f X ( x ).

x

f X( x ) = 20 x2 y3 dy = 5 x 4, 0

0 < x < 1.

g) Find E ( X ).

E ( X )

=

1

x 5 x 4 dx

0

=

5 6

.

h) Find f Y ( y ).

fY( y ) =

1

20 x2 y3 dx

y2

=

20 3

y3 y9

,

0 < y < 1.

i) Find E ( Y ).

E( Y ) =

1

0

y

20 3

y3 y9

dy

=

1

0

20 3

y

4

20 3

y 10

dy

=

4 3

20 33

=

8 11

.

j) Find E ( X Y ).

E(XY) =

1

0

x

x

0

y

20

x

2

y

3

dy

dx

=

1

4 x 11 2 dx

0

=

8 13

.

k) Find Cov ( X, Y ).

Cov ( X, Y )

=

E(XY) ? E(X)E(Y)

=

8 13

5 6

8 11

=

8 858

0.009324.

l) Are X and Y independent?

f ( x, y ) f X ( x ) f Y ( y ). X and Y are NOT independent. The support of ( X, Y ) is NOT a rectangle. X and Y are NOT independent. Cov ( X, Y ) 0. X and Y are NOT independent.

2. Let X and Y have the joint

probability density function

f X , Y ( x, y ) =

1,

x

x > 1, 0 < y < 1 , x

zero elsewhere.

a) Find f X ( x ).

1

fX(x) =

x

0

1

x

dy

=

1,

x2

x > 1.

b) Find E ( X ).

Since

1

x

1

x2

dx

=

1

1

x

dx

=

ln x 1

diverges,

E ( X ) does not exist.

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