PROBLEM 7E

Let the joint pmf of X and Y be

(a) Are X and Y independent?

(b) Calculate Cov(X,Y) and ρ.

This exercise also illustrates the fact that dependent random variables can have a correlation coefficient of zero.

Answer

Step 1 of 4

The given pmf is (x,y)S={(0,0), (1,1), (1,-1), (2,0)}

Arrange the given values into tabular form

0 |
1 |
2 |
||

-1 |
0 |
1/4 |
0 |
1/4 |

0 |
1/4 |
0 |
1/4 |
2/4 |

1 |
0 |
1/4 |
0 |
1/4 |

1/4 |
2/4 |
1/4 |
1 |

Step 2 of 4

a) If X and Y are independent then it satisfies E(XY)=E(X)E(Y)

E(X)=

= 0(1/4)+1(2/4)+2(1/4)

= 4/4

=1

E(Y)=

= -1(1/4)+0(2/4)+1(1/4)

= 1/4-1/4

=0