15.5.14 Two-dimensional random variables and their characteristics

Problem: The components of the two-dimensional discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) are independent random variables. The distribution tables of these random variables are following ones: \begin{tabular}{|c|c|c|} \hline\( x_{i} \) & -2 & 3 \\ \hline\( p_{i} \) & \( 5 / 6 \) & \( 1 / 6 \) \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\( y_{j} \) & -2 & -1 & 0 & 1 \\ \hline\( q_{j} \) & \( 1 / 4 \) & \( 1 / 4 \) & \( 1 / 4 \) & \( 1 / 4 \) \\ \hline \end{tabular} 1. Make the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \). 2.Calculate the expected value of this vector \( E[\eta] \) and the covariance matrix of \( V_{\eta} \). 3. Is it possible to immediately say what all the conditional series of distributions of each random variable will be?