Hey I want to check my solutions for this problem, can someone help me?
The random vector $X = (X_1, X_2)$ is uniformly distributed on the ellipse $S = {(x_1, x_2) ∈ \mathbb{R}^2 : x_1^2 + \frac{1}{4} x_2^2 \leq 1 }$ with an area of $2π$.
a. Provide the two-dimensional density of $X$. Show that $X_1$ and $X_2$ have the following densities for $z ∈ \mathbb{R}$: $f_1(z) = \frac{2}{π} \sqrt{1 − z^2 }1_{[−1,1]}(z), \ f_2(z) = \frac{2}{π} \sqrt{4 − z^2} 1_{[−2,2]}(z)$.
b. Are $X_1$ and $X_2$ uncorrelated? Are $X_1$ and $X_2$ independent?
I have done part a). For part b), I have shown that (since $f(x_1,x_2)=\frac{1}{2π}1_{x_1^2 + \frac{1}{4} x_2^2 \leq 1}$), $f(x_1,x_2)\neq f(x_1)f(x_2)$ does not hold for $x_1=1/2, x_2=1$, so $X_1$ and $X_2$ are not independent. To show uncorrelatedness, I need to demonstrate that $E[X_1X_2]=E[X_1]E[X_2]$.
For $E[X_1]= \frac{2}{π} \int_{-1}^{1} x_1\sqrt{1 − x_1^2 }=0$ Thus $E[X_1]E[X_2]=0$
For $E[X_1X_2]= 1/2\pi\int_{0}^{1}x_1 (\int_{-2\sqrt{1-x_1^2}}^{2\sqrt{1-x_1^2}}x_2 dx_2)dx_1=0$.
Therefore, we have that $E[X_1X_2]=E[X_1]E[X_2]$ and that the random variables are uncorrelated. Did I make any mistakes?