15.5.19 Two-dimensional random variables and their characteristics

Problem: Let's find the constant \( c \). Calculate the correlation coefficient \( \rho_{\xi_{1} \xi_{2}} \) and the probabilities of the following random events \( P\left\{\left(\xi_{1}=-2\right),\left(\xi_{2}<0\right)\right\}, P\left\{\xi_{2}>-1\right\} \), if the distribution table of a two-dimensional discrete random vector composed of these values has the form: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & -2 & -1 & 3 \\ \( x_{i} \) & & & \\ \hline-2 & 0.1 & 0.15 & \( c \) \\ \hline-1 & 0.15 & 0.25 & 0.15 \\ \hline \end{tabular}