15.5.17 Two-dimensional random variables and their characteristics

Problem: The ball is tossed into the basketball hoop. The probability of a success by one toss is equal to 0,85 . Let the random variable \( \xi_{1} \) be the number of successful tosses, and the random variable \( \xi_{2} \) be the number of misses by three tosses. 1. Make the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \). 2. Calculate the expected value of the random vector \( E[\eta] \) and its covariance matrix \( V_{\eta} \). 3. Find all the conditional distribution series for the random variable \( \xi_{1} \) under the condition that the random variable \( \xi_{2}=y_{j} \). 4. Calculate \( E\left[\xi_{2} / \xi_{1}=x_{i}\right] \). Draw all points of the form \( \left(x_{i} ; E\left[\xi_{2} / \xi_{1}=x_{i}\right]\right) \) in the Cartesian coordinate system. What can you say about the character of the distribution of these points?