Hello everyone,

I have this problem I can't quite figure out. I am considering the sample space $\displaystyle S$ being the unit circle without the boundary for $\displaystyle x > 0$.

The usual measure $\displaystyle \mu(\: \cdot \: )$ and $\displaystyle \sigma-algebra$ F. Probability is defined as being proportional the area of $\displaystyle A \in F$.

I am then to find the joint CDF for $\displaystyle R_1(x,y) = x \leq c_1$ and $\displaystyle R_2(x,y) = y \leq c_2$ for $\displaystyle c_1 \in (0,1)$ and $\displaystyle b \in (-1,1)$

This leads me to the following:

$\displaystyle P(R_1 \leq c_1 \land R_2 \leq b) = F_{R_1 R_2} (c_1,c_2)= \frac{\mu(A_{R_1 R_2})}{\mu(S)} = \frac{2}{\pi} \begin{cases} \int \limits_{-1}^{-\sqrt{1-c_1^2}} \int \limits_{0}^{\sqrt{1-y^2}} 1 dxdy + \int \limits_{0}^{c_1} \int \limits_{-\sqrt{1-c_1^2}}^{c_2} 1 dydx & c_2 > - \sqrt{1- {c_1}^2} \\ \frac{1}{2} + \frac{c_2 \sqrt{1-c_2^2}}{\pi} + \frac{\sin^{-1}(c_2)}{\pi}& otherwise \end{cases} $

However, I am not sure how to very I find the correct distribution. And it seems odd that the area are give by such messy expressions.

Kind Regards