Find variable that is uncorrelated but not independent

Question

I am given PMF of random variable X. P(X=0) = P(x=1) =0.5. Now there is another RV Y such that Y = XZ. I have to find Z independent of X such that X and Y are uncorrelated but not independent. My first intuition is that the PMF of Z should be 1/z^3. And Z can take values form 1 to n. This way each time it is multiplied with X, U gets reduced. So, even if X is increased to 1 U won't increase. Is this correct?

Answer 1

You've mentioned both U and Y as RVs. I'll assume that they both refer to the same variable so U=XZ.

Note that Cov(U,X)=0 E(UX) - E(U)E(X)

Since Z and X are independent we have E(U) = E(XZ) = E(X).E(Z)

Substituting that in the equation above we get:

= E(X^2).E(Z) - E(Z).E(X)^2

=E(Z).(E(X^2) - E(X)^2)

=E(Z).var(X)

Now we know that var(X) is not 0. So just define Z such that E(Z) = 0.

That's pretty simple to do. Just define an RV symmetric around 0. So +-1 both with probability 0.5 and you get your Z.