PROBLEM 6E
The joint pmf of X and Y is f (x, y) = 1/6, 0 ≤x + y ≤ 2, where x and y are nonnegative integers.
(a) Sketch the support of X and Y.
(b) Record the marginal pmfs fX(x) and fY(y) in the “margins.”
(c) Compute Cov(X,Y).
(d) Determine ρ, the correlation coefficient.
(e) Find the bestfitting line and draw it on your figure.
Answer
Step 1 of 8
a) Construct a table for a given pmf 0
y x 
0 
1 
2 

0 
1/6 
1/6 
1/6 
3/6 
1 
1/6 
1/6 
0 
2/6 
2 
1/6 
0 
0 
1/6 
3/6 
2/6 
1/6 
1 
Step 2 of 8
b) Marginal pmf of x is
x 
0 
1 
2 
Total 
3/6 
2/6 
1/6 
1 
Marginal pmf of y is
y 
0 
1 
2 
Total 
3/6 
2/6 
1/6 
1 
Step 3 of 8
c) Now find the mean and variance by using the tables
E(X) =
=0(3/6)+1(2/6)+2(1/6)
=4/6
E(Y)=
=0(3/6)+1(2/6)+2(1/6)
=4/6
E(XY)=
=1(1)(⅙)
=1/6
cov(X,Y)=E(XY)E(X)E(Y)
=1/64/6(4/6)
= 10/36
Step 4 of 8
d)E(X2)=
=0(3/6)+1(2/6)+4(1/6)
=1
=V(x)
=E(X2) [ E(X)]2
=1 (4/6)2
=20/36