Posted: December 21st, 2015

# Find the marginal distribution of X .

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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