Two-dimensional random variable $ p_{\xi_1\xi_2}(x_1,x_2)=\begin{cases} \frac 1 {6\pi},& \frac {{x_1}^2} 9+\frac {{x_2}^2} 4 \leqslant 1,\\ 0,&\frac {{x_1}^2} 9+\frac {{x_2}^2} 4 > 1. \end{cases} $
Are two random variables $\xi_1$ and $\xi_2$ independent?
I know if they are independent then $p_{\xi_1\xi_2}(x_1,x_2)=p_{\xi_1}(x_1)p_{\xi_2}(x_2)$. But how I can use that?
Compute the marginal pdfs $$ p_{\xi_1}(x_1)=\frac{1}{6\pi}\int_{-\frac{1}{3} \sqrt{36-4 x_1^2}}^{\frac{1}{3} \sqrt{36-4 x_1^2}}dx_2=\frac{1}{9\pi} \sqrt{36-4 x_1^2}\qquad -3\leq x_1\leq 3 $$ and $$ p_{\xi_2}(x_2)=\frac{1}{6\pi}\int_{-\frac{1}{2} \sqrt{36-9 x_2^2}}^{\frac{1}{2} \sqrt{36-9 x_2^2}}dx_1=\frac{1}{6\pi}\sqrt{36-9 x_2^2}\qquad -2\leq x_2\leq 2\ . $$ The product of the two does not reproduce $p_{\xi_1,\xi_2}(x_1,x_2)$, so the two variables are not independent.
