I need answers to these questions

MATH 240 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

yxyx 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).