15.2.24 One dimensional random variables and their characteristics

Problem: The distribution table of the discrete random variable \( \xi \) has the form: \begin{tabular}{|c|c|c|c|c|c|} \hline\( x_{i} \) & \( -\pi / 2 \) & \( -\pi / 6 \) & \( \pi / 2 \) & \( \pi / 6 \) & \( \pi \) \\ \hline\( p_{i} \) & 0.3 & 0.2 & 0.1 & 0.1 & 0.3 \\ \hline \end{tabular} 1. Make the distribution tables and find the distribution functions for the random variables \( \tau_{i} \), \( i=1,2,3 \), if: a) \( \tau_{1}=\cos \xi \) b) \( \tau_{2}=\sin \xi \) c) \( \tau_{3}=|\sin \xi| \). 2. Calculate the expected values \( E\left[\tau_{i}\right] \) and the dispersions \( V\left[\tau_{i}\right], i=1,2,3 \).