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?
Prove these two variables are dependent.
0
$\begingroup$
probability
probability-theory
-
3$x^2+y^2=1$,thus they are not independent – 2012-10-19
1 Answers
2
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 $y$.
-
0@Dilip: You're right; fixed. – 2012-10-19