15.5.8 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 known: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & 0 & 1 & 3 \\ \hline\( x_{i} \) & & & \\ \hline-2 & 0.01 & 0.02 & 0.03 \\ \hline-1 & 0.01 & 0 & 0.01 \\ \hline 0 & 0.02 & 0.5 & 0.4 \\ \hline \end{tabular} 1) Find the marginal laws of distribution for random variables. 2) 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 \( \xi_{1}=-1 \). Will the random variables \( \xi_{1} \) and \( \xi_{2} \) be dependent? 3) Calculate \( E\left[\xi_{1}^{2}-\xi_{2}\right] \). 4) Find the probability \( P\left\{\left(\xi_{1}^{2}-\xi_{2}\right)=-1\right\} \).