15.5.4 Two-dimensional random variables and their characteristics

Problem: The distribution table of the discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) is given: \begin{tabular}{|c|c|c|} \hline\( x_{i} y_{j} \) & 0 & 1 \\ \hline-1 & 0.17 & 0.1 \\ \hline 0 & 0.13 & 0.3 \\ \hline 1 & 0.25 & \( ? \) \\ \hline \end{tabular} 1) Find the marginal (partial) distribution tables for random variables \( \xi_{1} \) and \( \xi_{2} \). 2) Calculate \( E\left[\xi_{1}\right], E\left[\xi_{2}\right], V\left[\xi_{1}\right], V\left[\xi_{2}\right] \), as well as the moment of correlation \( V_{\xi_{1} \xi_{2}} \) and the correlation coefficient \( \rho_{\xi 1 \xi_{2}} \). 3) Compose a conditional distribution series of the random variable \( \xi_{1} \) under the condition that the random variable \( \xi_{2}=1 \), and then a conditional distribution series of the random variable \( \xi_{2} \) under the condition that the random variable \( \xi_{1}=0 \). Will the random variables \( \xi_{1} \) and \( \xi_{2} \) be independent? 4) Calculate the values of conditional expected values \( E\left[\xi_{1} / \xi_{2}=1\right] \) and \( E\left[\xi_{2} / \xi_{1}=0\right] \).