I need answers to these questions

MATH240 HOMEWORK IV

(Due: next class time for each section)

1. The density function of X is

 






 



otherwise

xbxa
xf x

0

102

(1)

If E[X] = 3/5, find a and b. What is Var[X]?

2. Given the joint PMF of X and Y as

(a) Find E[X], E[Y], and E[XY],
(b) Show that X and Y are uncorrelated,
(c) Determine if X and Y are independent or not.

3. The joint density function of X and Y is

 






 


otherwise

yx

yx

yxf yx

0

10,10
,,

(2)

a) Are X and Y independent?
b) Find the probability density function of X?
c) Find P(X+Y < 1)=?

4. Two fair dice are rolled. Find the joint probability mass function of X and
Y when X is the largest value obtained on any die and Y is the sum of the
values.

5. Suppose the joint density function of X and Y is given by

 

















otherwise

yx
y

ee
yxf

yyx

yx
0

0,0
,

/

, (3)

a) Find the conditional probability density of X, given that Y=y.
b) Find P(X > 1 | Y = y).