STAT 462 Homework 7

STAT 462

Homework 7 due Thursday, November 1, 2012

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 Yf x y x y x y    

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

, ( , ) = 2 , 0 1, 0 1.X Yf x y x x y   

3. Define

2 2

,

21 0 < < < 1, 0

( , ) = 2 0

X Y

x y x y x f x y

otherwise

 

 

Find the marginal distributions of X and Y .

4. Let 1Y and 2Y 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 1Y and 2Y is

modeled by the pdf

1 2

1 2 1 2 , 1 2

0 1, 0 1 ( , ) = .

0 Y Y

y y y y f y y

otherwise

      

(a) Find  1 21 1, .2 4P Y Y  (b) Find  1 2 1 .P Y Y  (c) Find 1 2( 1 / 2 | 1 / 2)P Y Y  . (d) Find the marginal probability density functions for 1Y and 2.Y

(e) Are 1Y and 2Y 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 1Y + 25 2Y . 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 XYf given by

   

1 1

1 0,1,.... , 0 ,1 0, 0!

, .

0

x y

XY

y x x e

y f x y

else

     

            

      

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

(b) Calculate the marginal pmf ( ).Yf y

• Book Problems: 5.15, 5.52, 5.59