Suppose $\theta$ is a Uniform random variable on [0, 2$\pi$]. Let $X$ be $cos(\theta)$ and $Y$ be $sin(\theta)$. We have to show that $X$ and $Y$ are uncorrelated but not independent.
My solution:
$f_{\theta}(\theta) = \frac{1}{2\pi} $ , $0 < \theta < 2\pi$
Using LOTUS,
$E[X] = E[cos(\theta)] = \int_{0}^{2\pi} cos(\theta)\frac{1}{2\pi}d\theta$ = $0$
Similarly,
$E[Y] = E[sin(\theta)] = \int_{0}^{2\pi} sin(\theta)\frac{1}{2\pi}d\theta$ = $0$
$XY$ = $\frac{sin(2\theta)}{2}$. Therefore,
$E[XY] = \int_{0}^{2\pi} \frac{sin(2\theta)}{2}\frac{1}{2\pi}d\theta$ = $0$
Therefore $Cov(X,Y) = E[XY] - E[X]E[Y]$ = 0.
Correlation $\rho$ = $0$.
My question is:
Since $E[XY] = E[X]E[Y] (0 = 0 \cdot 0)$, wouldn't it also mean that they are independent. What am I doing wrong here?
These random variables are indeed uncorrelated but not independent, there is nothing wrong - zero correlation doesn't imply independence. To see that this variables are dependent you can check, for example, that $P(X > 0.9) > 0$, $P(Y > 0.9) > 0$, $P(X > 0.9, Y > 0.9) = 0$.