Finding joint CDF from joint PDF.

Question

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.

Answer 1

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