Microsoft Word - homework7
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,
espectively, 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
( , ) = .
0Y 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