I know that if 2 random variables are indepedent, then E(XY)=E(X)E(Y)
IN such a case COv(XY)=E(XY)-E(X)E(Y)=0
But if Cov(XY)=0 then X and Y may or may not be independent.
But in given example, how to say 2 random variables U and V are independent or not
X~ random variable and $f(x)= 1+x, -1 <x \leq 0 ; \\ 1-x , 0<x<1, \\ 0 ,otherwise$
To check whether U=X and $V=X^2$ are independent or not
My attempt:
$E(x)=0 = E(x^3)$
$E(x^2)=1/6$
$E(UV)=E(x^3)=0$
$E(x)E(X^2)=0$ and also $Cov(U,V)=0$
My doubt is whether U and V are independent or not as i am getting E(UV)-E(U)E(V)=0 and also Cov(U,V)=0
If zero covariance does not imply independence, then why are you trying to calculate the covariance to show independence? That won't tell you anything useful. It's like saying that showing a number $x$ is positive does not imply that it is even, but then you are trying to prove $x$ is positive and wondering why this doesn't prove $x$ is even.
Instead, $U = X$ and $V = X^2$ clearly cannot be independent, because knowledge of $X$ immediately tells you the value of $X^2$. This is trivial.