<s>[INST] Finding joint CDF from joint PDF. 
Suppose $X$ and $Y$ are random variables with joint density 
  $$
f(x,y) = \left\{
        \begin{array}{ll}
            1/\pi & \quad\text{if}\ x^{2} + y^{2} \leq 1 \\
            0 & \quad \mathrm {otherwise}
        \end{array}
    \right.$$
  Find the joint CDF of $X$ and $Y$.

How do I proceed? Please help me as I am a beginner of this topic.
 [/INST] I think the joint CDF $F(x,y)= \mathbb P(X \le x,Y \le Y)$ may be fairly involved, and looking at this diagram suggests the following to me:



*

*If $x \le -1$, or if $y \le -1$, or if $x \le 0$ and $y \le 0$ and $x^2+y^2 \ge 1$:


$$F(x,y)=0$$ 


*

*If $x \ge 1$ and $y \ge 1$:


$$F(x,y)=1$$ 


*

*If  $-1 \le x \le 1$ and $y \ge 1$, or if $-1 \le x \le 0$ and $0 \le y \le 1$ and $x^2+y^2 \ge 1$:


$$F(x,y)= \frac1\pi\left( \frac\pi 2 +  \sin^{-1}(x) +  x\sqrt{1-x^2} \right)$$ 


*

*If  $x \ge 1$ and  $-1 \le y \le 1$, or if $0 \le x \le 1$ and  $-1 \le y \le 0$ and $x^2+y^2 \ge 1$:


$$F(x,y)= \frac1\pi\left( \frac\pi 2 +  \sin^{-1}(y) +  y\sqrt{1-y^2} \right)$$  


*

*If  $0 \le x \le 1$ and  $0 \le y \le 1$ and $x^2+y^2 \ge 1$:


$$F(x,y)= \frac1\pi\left( \sin^{-1}(x)+  \sin^{-1}(y) +  x\sqrt{1-x^2}  + y\sqrt{1-y^2} \right)$$ 


*

*If  $x^2+y^2 \le 1$:


$$F(x,y)= \frac1\pi\left( \frac\pi 4 + \frac1{2}\sin^{-1}(x) + \frac1{2} \sin^{-1}(y) +  \frac1{2} x \sqrt{1-x^2} + \frac1{2}y \sqrt{1-y^2} + x y \right)$$ 
The interesting parts of the marginal CDFs for $X$ and $Y$ are in the third and fourth points above respectively 
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