15.5.16 Two-dimensional random variables and their characteristics

Problem: Two tetrahedrons, with numbered faces, are tossed. Let \( \xi_{1} \) be the random variable, which takes values equal to 1 , if the sum of the numbers on the lower faces is an odd number, and the value is equal to 0 , if the indicated sum will be an odd number. Let \( \xi_{2} \) be a random variable, which is equal to 1 , if the sum of the points on the lower faces of both tetrahedrons is divisible by 4 , and is otherwise equal to 0 . 1. Find the distribution table of the two-dimensional discrete random vector \( \eta \), composed of these variables. 2. Calculate the expected value of the vector \( E[\eta] \) and the covariance matrix \( V_{\eta} \). 3. Find all the conditional distribution series for the random variable \( \xi_{1} \) under the condition that the random variable \( \xi_{2}=y_{j} \), and then all the conditional distribution series for the random variable \( \xi_{2} \) under the condition that the random variable \( \xi_{1}=x_{i} \). Will the random variables \( \xi_{1} \) and \( \xi_{2} \) be dependent? 4. Calculate all the values of the conditional expected values \( E\left[\xi_{1} / \xi_{2}\right] \) and \( E\left[\xi_{2} / \xi_{1}\right] \).