I am trying to solve the following question.
Random Variable X has the following PMF:
P(X = 0) = 0.5
P(X = 1) = 0.5
We define another random variable U = XZ. Construct random variable Z , independent of X such that Cov(U,X) = 0 i.e. U and X are uncorrelated but not independent.
As we can see that Z = U/X, U and X are also dependent then how can we construct a random variable Z which is independent of X. One way i was thinking is that write it in term of U but in turn U is also dependent on X.
How to approach this problem
${Cov(UX)=E(UX)-E(U)E(X)={E(X^2)E(Z)-E(Z)(E(X))^2,}}$
since $X$ and $Z$ are independent. For $Cov(UX)=0$, either $E(Z)=0$ or $E(X^2)-(E(X))^2=0$. We know the latter cannot be $0$, since it is the variance of $X$. To make $E(Z)=0$, let $Z$ be any random variable symmetric around $0$, for example $Z=\pm 1$, each with probability $0.5$.