Find an example of two random variables X1 and X2 which are

Find an example of two random variables X1 and X2, which are dependent but uncorrelated. You need to prove that your choice of X1 and X2 are dependent but uncorrelated.

Solution

Let X be a random variable with the value 0 with probability 1/2, and takes the value 1 with probability ½

Thus, E(U)=E(XZ)=E(X)E(Z)=0

Thus, Cov(U, X)=0

Thus, the two variables are dependent but not correlated.