Posted: December 21st, 2015

2

1. A pdf is defined by

( 2 ) if 0 < <1 and 0 < < 2

( , )=

0 otherwise

C x y y x

f x y

(a) Find the value of C .

(b) Find the marginal distribution of X .

(c) Find the joint cdf of X and Y . Define for all .

2. (a) Find P(X > Y ) if X and Y are jointly distributed with pdf

, ( , )= , 0 1, 0 1. X Y f x y x y x y

(b) Find P(X 2 < Y < X ) if X and Y are jointly distributed with pdf

, ( , )=2 , 0 1, 0 1. X Y f x y x x y

3. Define

2 2

,

21 0 < < <1, 0

( , )= 2

0

X Y

x y x y x

f xy

otherwise

Find the marginal distributions of X and Y .

4. Let 1 Y and 2 Y denote the proportion of time (out of one workday) during which employees I and II,

respectively, perform their assigned tasks. The joint relative frequency behavior of 1 Y and 2 Y is

modeled by the pdf

1 2

1 2 1 2

, 1 2

0 1,0 1

( , )= .

Y Y 0

y y y y

f y y

otherwise

(a) Find 1 2

1 , 1 . P Y 2 Y 4

(b) Find 1 2 P Y Y 1 .

(c) Find 1 2 P(Y 1/ 2 |Y 1/ 2) .

(d) Find the marginal probability density functions for 1 Y and 2Y .

(e) Are 1 Y and 2 Y independent? Why or why not?

(f) Employee I has a higher productivity rating than Employee II and a measure of the total productivity

of the pair of employees is 30 1 Y + 25 2 Y . Find the expected value of this measure of productivity.

(g) Find the variance for the measure of productivity in (f).

5. The random variables X and Y have the joint distribution XY f given by

1 1

1 1 0,1,…. , 0 ,

! 0, 0

, .

0

x

y

XY

y x

x e

y

f xy

else

(a) Calculate the marginal pdf ( ). X f x Identify this distribution and its parameter(s).

(b) Calculate the marginal pmf ( ). Y f y

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