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Problem: A «wrong» coin (the probability of getting "heads" is \( A \) ) is tossed for \( N \) times. The following variables are considered: \( x \) is the number of "heads", \( y \) is the number of "tails", \[ z_{1}=\frac{x}{y}, z_{2}=x+y, z_{3}=\frac{x}{z_{2}} \text {. } \] Answer the following questions about these random variables: a) describe the distributions of the random variables \( x, y, z_{1}, z_{2}, z_{3} \); find the expected values, the second moments, dispersions; b) describe the conditional distribution of the random variables \( x \mid y \); c) as a result of the \( M \)-th toss, it turned out that there are exactly \( L \) «heads», what is the probability that there will be no more than \( K \) "tails"? d) Let's find the covariance and the correlation coefficient of the variables \( x \) and \( y \); e) Find the covariance and the correlation coefficient of the variables \( x^{2} \) and \( y \); \[ A=0,69 ; N=252 ; M=142 ; L=80 ; K=55 \text {. } \]

15.5.2 Two-dimensional random variables and their characteristics

12.85 $

Problem: The distribution table of the discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) is given: \begin{tabular}{|c|c|c|} \hline\( x_{i} y_{j} \) & 0 & 1 \\ \hline-1 & 0.17 & 0.1 \\ \hline 0 & 0.13 & 0.3 \\ \hline 1 & 0.25 & \( ? \) \\ \hline \end{tabular} 1) Find the marginal (partial) distribution tables for random variables \( \xi_{1} \) and \( \xi_{2} \). 2) Calculate \( E\left[\xi_{1}\right], E\left[\xi_{2}\right], V\left[\xi_{1}\right], V\left[\xi_{2}\right] \), as well as the moment of correlation \( V_{\xi_{1} \xi_{2}} \) and the correlation coefficient \( \rho_{\xi 1 \xi_{2}} \). 3) 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 independent? 4) Calculate the values of conditional expected values \( E\left[\xi_{1} / \xi_{2}=1\right] \) and \( E\left[\xi_{2} / \xi_{1}=0\right] \).

15.5.4 Two-dimensional random variables and their characteristics

6.42 $

Problem: The discrete random variables \( \xi_{1} \) and \( \xi_{2} \) are independent and have the following distribution tables: \( \xi_{1} \) \begin{tabular}{|c|c|c|c|} \hline\( x_{i} \) & 0 & 1 & 3 \\ \hline\( p_{i} \) & \( 1 / 2 \) & \( 3 / 2 \) & \( 1 / 8 \) \\ \hline \end{tabular} \[ \xi_{2} \] \begin{tabular}{|c|c|c|} \hline\( y_{j} \) & 0 & 1 \\ \hline\( q_{j} \) & \( 1 / 3 \) & \( 2 / 3 \) \\ \hline \end{tabular} 1) Find the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \), composed of these variables. 2) Calculate the expected value and the dispersion of each random variable. 3) Calculate the correlation moment \( V_{\xi_{1} \xi_{2}} \), and then find \( E\left[\xi_{1}, \xi_{2}\right] \). Will these variables be correlated? Would it be possible, without calculating \( V_{\xi_{1} \xi_{2}} \), to immediately assume what it is equal to?

15.5.5 Two-dimensional random variables and their characteristics

2.57 $

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?

15.5.7 Two-dimensional random variables and their characteristics

5.14 $

Problem: The distribution table of the discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) is known: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & 0 & 1 & 3 \\ \hline\( x_{i} \) & & & \\ \hline-2 & 0.01 & 0.02 & 0.03 \\ \hline-1 & 0.01 & 0 & 0.01 \\ \hline 0 & 0.02 & 0.5 & 0.4 \\ \hline \end{tabular} 1) Find the marginal laws of distribution for random variables. 2) 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 \( \xi_{1}=-1 \). Will the random variables \( \xi_{1} \) and \( \xi_{2} \) be dependent? 3) Calculate \( E\left[\xi_{1}^{2}-\xi_{2}\right] \). 4) Find the probability \( P\left\{\left(\xi_{1}^{2}-\xi_{2}\right)=-1\right\} \).

15.5.8 Two-dimensional random variables and their characteristics

3.34 $

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?

15.5.10 Two-dimensional random variables and their characteristics

3.85 $

Problem: The joint probability distribution of the change in the yield of two stocks of the form \( A \) and \( B \) (random variables \( \xi_{1} \) and \( \xi_{2} \) ) is given by the following table: \begin{tabular}{|c|c|c|c|c|c|} \hline\( y_{j} \) & \( 2 \% \) & \( 6 \% \) & \( 9 \% \) & \( 15 \% \) & \( 20 \% \) \\ \hline \( 6 \% \) & 0.1 & 0 & 0 & 0 & 0 \\ \hline \( 8 \% \) & 0 & 0.2 & 0 & 0 & 0 \\ \hline \( 10 \% \) & 0 & 0 & 0.4 & 0 & 0 \\ \hline \( 12 \% \) & 0 & 0 & 0 & 0.2 & 0 \\ \hline \( 14 \% \) & 0 & 0 & 0 & 0 & 0.1 \\ \hline \end{tabular} 1. Compose partial distribution laws for each indicator (for random variables \( \xi_{1} \) and \( \xi_{2} \) ). 2. Find the expected yield and the yield curve risk for each valuable security (calculate the expected value and the dispersion for each random variable). 3. Do the yields of valuable securities correlate with each other? What is the power of dependence between them (calculate the correlation coefficient)? 4. Find a conditional distribution series for the yields of the stock of the form \( A \) (the first random variable \( \xi_{1} \) ) under the condition that the yields of the stock of the form \( B \) is equal to \( 9 \% \) (the random variable \( \xi_{2}=9 \) ), and then a conditional distribution series for the yields of the stock of the form \( B \) (the random variable \( \xi_{2} \) ) under the condition that the yields of the stock of the form \( A \) is equal to \( 14 \% \) (the random variable \( \xi_{1}=14 \) ). Will the yields of these valuable securities (random variables \( \xi_{1} \) and \( \xi_{2} \) ) dependent?

15.5.13 Two-dimensional random variables and their characteristics

3.85 $

Problem: The components of the two-dimensional discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) are independent random variables. The distribution tables of these random variables are following ones: \begin{tabular}{|c|c|c|} \hline\( x_{i} \) & -2 & 3 \\ \hline\( p_{i} \) & \( 5 / 6 \) & \( 1 / 6 \) \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\( y_{j} \) & -2 & -1 & 0 & 1 \\ \hline\( q_{j} \) & \( 1 / 4 \) & \( 1 / 4 \) & \( 1 / 4 \) & \( 1 / 4 \) \\ \hline \end{tabular} 1. Make the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \). 2.Calculate the expected value of this vector \( E[\eta] \) and the covariance matrix of \( V_{\eta} \). 3. Is it possible to immediately say what all the conditional series of distributions of each random variable will be?

15.5.14 Two-dimensional random variables and their characteristics

5.14 $

Problem: The distribution table of the two-dimensional discrete random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \) is given: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & -2 & 0 & 2 \\ \hline\( x_{i} \) & & & \\ \hline-3 & \( 1 / 8 \) & \( \mathrm{c} \) & \( 7 / 24 \) \\ \hline 3 & \( 5 / 24 \) & \( 1 / 6 \) & \( 1 / 8 \) \\ \hline \end{tabular} 1. Find the constant \( c \) and partial distribution laws of the random variables \( \xi_{1} \) and \( \xi_{2} \). 2. Calculate the expected values \( E\left[\xi_{1}\right], E\left[\xi_{2}\right] \) and the dispersions \( V\left[\xi_{1}\right], V\left[\xi_{2}\right] \), as well as the moment of correlation \( V_{\xi_{1} \xi_{2}} \) and the correlation coefficient \( \rho_{\xi_{1} \xi_{2}} \). 3. Will the random variables be independent? 4. Make the distribution table of the new random vector \( \eta_{1}=\left(\tau_{1}, \tau_{2}\right)^{T} \), where \( \tau_{1}=\xi_{1}+\xi_{2}, \tau_{2}=\xi_{1} \xi_{2} \). 5. Find \( E\left[\eta_{1}\right] \) and the covariance matrix \( V_{\eta_{1}} \).

15.5.15 Two-dimensional random variables and their characteristics

6.42 $