Given two random variables X = cos(θ) and Y = sin(θ), where θ is uniformly distributed over [0,π], they are not independent. How can I prove this non-independence rigorously?

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    $\begingroup$ $x^2+y^2=1$,thus they are not independent $\endgroup$ – Golbez Oct 19 '12 at 13:06

You can deduce the value of $Y$ from that of $X$, $y=\sqrt{1-x^2}$. Thus you have full knowledge of $y$ given $x$, whereas independence implies that knowing $x$ gives you no knowledge at all about $y$.

  • $\begingroup$ Isn't it the other way around? Given that the value of $X$ is $x \in [-1,+1]$, $Y$ has unique value $\sqrt{1-x^2} \geq 0$ while given that the value of $Y$ is $y \in [0,1]$, we still have some uncertainty about the value of $X$ which can be $\pm \sqrt{1-y^2}$? Of course, the conditional distribution of $X$ given $Y$ does depend on the value of $Y$, and this suffices to assert that $X$ and $Y$ are dependent. $\endgroup$ – Dilip Sarwate Oct 19 '12 at 15:08
  • $\begingroup$ @Dilip: You're right; fixed. $\endgroup$ – joriki Oct 19 '12 at 15:20

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