15.5.10 Two-dimensional random variables and their characteristics

Problem: Two cubes are tossed. Let \( \xi_{1} \) be the random variable which takes the values, equal to 1 , if the sum of the numbers on the upper faces of both cubes is an even number, and the values equal to 0 , if the indicated sum is an odd number. Let \( \xi_{2} \) be a random variable, which is equal to 1 , if the sum of the numbers of the upper faces of both cubes is divided by 3 , and is otherwise equal to 0 . 1. Find the distribution table of the two-dimensional random discrete vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \), composed of these variables. 2. Calculate the expected value of this vector \( E[\eta] \) and the covariance matrix \( V_{\eta} \). 3. Will these random variables be correlated? 4. Will these random variables be dependent?