I'm learning statistics and trying to solve the following problem:
Continuous random variable $X$ ~ $R(0,2)$. Describe the dependency of $Y$ on $X$ for $Y=X^2$. Write a covariance matrix and correlation matrix of a random vector $(X,Y)$.
I calculated the expected value of $X$, which is $1$. I also calculated the dispersion, which is $1/3$. Not sure where to go from now. Could you please help me?
You are on the right track. Just apply the definitions.
Correlation matrix: $ R=E\left\{\begin{bmatrix} X \\ Y \end{bmatrix} [X, Y] \right\} = \begin{bmatrix} E\{X^2\} & E\{XY\} \\ E\{XY\} & E\{Y^2\} \end{bmatrix}$
Covariance matrix: $C = E\left\{\begin{bmatrix} \tilde{X} \\ \tilde{Y} \end{bmatrix} [\tilde{X}, \tilde{Y}] \right\} = \begin{bmatrix} E\{\tilde{X}^2\} & E\{\tilde{X}\tilde{Y}\} \\ E\{\tilde{X}\tilde{Y}\} & E\{\tilde{Y}^2\} \end{bmatrix}$
where $\tilde{X} = X-E\{X\}$.
Now, you have calculated $E\{X\}$ and and $E\{\tilde{X}^2\}$ correctly. You already have $E\{X^2\}$, since it equals $E\{\tilde{X}^2\}+(E\{X\})^2$. As for $E\{XY\}$, it equals $E\{X^3\}$. Also $E\{Y^2\}=E\{X^4\}$. And so on. For example,
$E\{Y\} = E\{X^2\} = \dfrac{1}{3}+1^2 = \dfrac{4}{3}$
$E\{XY\} = E\{X^3\}=\int\limits_{0}^{2} \dfrac{1}{2} x^3 dx = \dfrac{1}{2} \left[ \dfrac{1}{4} x^4\right]_{x=0}^{2} = 2 $
and so on.