Here are two random variables that are uncorrelated but not independent. Let X and Y have the following joint probability mass function:

x |
y |
p(x, y) |

-1 |
1 |
1/3 |

0 |
0 |
1/3 |

1 |
1 |
1/3 |

a. Use the definition of independence on page 141 to show that X and Y are not independent (in fact Y = |X |, so Y is actually a function of X).

b. Show that X and Y are uncorrelated.

Answer :

Step 1 of 3:

Given, let X and Y have the joint probability mass function.

x |
y |
p(x,y) |

-1 |
1 |
1/3 |

0 |
0 |
1/3 |

1 |
1 |
1/3 |

Step 2 of 3:

The claim is to show that X and Y are not independentIf X and Y are independent

Then, P(X = 1 and Y = 1 ) = P(X =1) P(Y=1)

From the above table we have

P(X = 1 and Y = 1 ) = ⅓, P(X = 1 ) = ⅓ and P( Y = 1) = ⅔

Then, P(X =1) P(Y=1) = (⅓) (⅔)

= 2/9

Therefore, P(X = 1 and Y = 1 ) P(X =1) P(Y=1)

Hence, X and Y are not independent.