15.5.5 Two-dimensional random variables and their characteristics

Problem: The discrete random variables \( \xi_{1} \) and \( \xi_{2} \) are independent and have the following distribution tables: \( \xi_{1} \) \begin{tabular}{|c|c|c|c|} \hline\( x_{i} \) & 0 & 1 & 3 \\ \hline\( p_{i} \) & \( 1 / 2 \) & \( 3 / 2 \) & \( 1 / 8 \) \\ \hline \end{tabular} \[ \xi_{2} \] \begin{tabular}{|c|c|c|} \hline\( y_{j} \) & 0 & 1 \\ \hline\( q_{j} \) & \( 1 / 3 \) & \( 2 / 3 \) \\ \hline \end{tabular} 1) Find the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \), composed of these variables. 2) Calculate the expected value and the dispersion of each random variable. 3) Calculate the correlation moment \( V_{\xi_{1} \xi_{2}} \), and then find \( E\left[\xi_{1}, \xi_{2}\right] \). Will these variables be correlated? Would it be possible, without calculating \( V_{\xi_{1} \xi_{2}} \), to immediately assume what it is equal to?