15.5.15 Two-dimensional random variables and their characteristics

Problem: The distribution table of the two-dimensional discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) is given: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & -2 & 0 & 2 \\ \hline\( x_{i} \) & & & \\ \hline-3 & \( 1 / 8 \) & \( \mathrm{c} \) & \( 7 / 24 \) \\ \hline 3 & \( 5 / 24 \) & \( 1 / 6 \) & \( 1 / 8 \) \\ \hline \end{tabular} 1. Find the constant \( c \) and partial distribution laws of the random variables \( \xi_{1} \) and \( \xi_{2} \). 2. Calculate the expected values \( E\left[\xi_{1}\right], E\left[\xi_{2}\right] \) and the dispersions \( V\left[\xi_{1}\right], V\left[\xi_{2}\right] \), as well as the moment of correlation \( V_{\xi_{1} \xi_{2}} \) and the correlation coefficient \( \rho_{\xi_{1} \xi_{2}} \). 3. Will the random variables be independent? 4. Make the distribution table of the new random vector \( \eta_{1}=\left(\tau_{1}, \tau_{2}\right)^{T} \), where \( \tau_{1}=\xi_{1}+\xi_{2}, \tau_{2}=\xi_{1} \xi_{2} \). 5. Find \( E\left[\eta_{1}\right] \) and the covariance matrix \( V_{\eta_{1}} \).