Finding CDF and PDF of Y from $X^2 + Y^2 = 1$

Question

$(X,Y)\in \{(x,y)\in \mathbb{R}^2: x^2+y^2 = 1\}$

Now is X uniformly distributed on $[-1, 1]$ and $Y \ge 0$

What I do not understand is how to get the CDF and PDF of Y. What steps should I follow to find them?

Answer 1

Find the interpretation of $Y\le t$ for any $t\in[0,1]$ : $$Y\le t\iff 1-X^2=Y^2\le t^2 \iff X^2\ge 1-t^2 \iff X\ge \sqrt{1-t^2}\text{ or } X\le -\sqrt{1-t^2}$$ Now you can compute the probability of event $Y\le t$ : $${\rm P}(Y\le t) = {\rm P}(X\in[-1,-\sqrt{1-t^2}]\cup[\sqrt{1-t^2},1]) = 1-\sqrt{1-t^2}$$ You then obtain PDF by taking derivative of this result.