Example 1
- Let be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2.
- Let be a random variable, independent of , that takes the value −1 with probability 1/2, and takes the value 1 with probability 1/2.
- Let be a random variable constructed as .
The claim is that and have zero covariance (and thus are uncorrelated), but are not independent.
Proof:
Taking into account that
where the second equality holds because and are independent, one gets
Therefore, and are uncorrelated.
Independence of and means that for all and , . This is not true, in particular, for and .
Thus so and are not independent.
Q.E.D.
Example 2
If is a continuous random variable uniformly distributed on and , then and are uncorrelated even though determines and a particular value of can be produced by only one or two values of :
on the other hand, is 0 on the triangle defined by although is not null on this domain.
Therefore and the variables are not independent.
Therefore the variables are uncorrelated.