15.5.7 Two-dimensional random variables and their characteristics

Problem: The batch of products contains 4 products with a defect and 6 proper-quality products. One product is taken out randomly, its quality is tested and it's not returned to the batch. From the products remaining in the batch another one is selected, tested. Let \( \xi_{1} \) be the number of proper-quality products during the first retreatment, and the random variable \( \xi_{2} \) is the number of proper-quality products during the first retreatment. 1. Make the joint distribution table of these random variables. 2. Find the marginal distribution laws of these variables. 3. Calculate the expected values of these random variables and write the expected value of the vector \( \eta==\left(\xi_{1}, \xi_{2}\right)^{T} \). 4. Calculate the dispersions, moment and the correlation coefficient of these random variables. Write the covariance and correlation matrices. 5. Compose a conditional distribution series of the random variable \( \xi_{1} \) under the condition that the random variable \( \xi_{2}=1 \), and then a conditional distribution series of the random variable \( \xi_{2} \), under the condition that the random variable \( \xi_{1}=0 \). Will the random variables \( \xi_{1} \) and \( \xi_{2} \) be dependent?