Two-dim. random variables and their characteristics

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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 \( E\left[\xi_{2} / \xi_{1}\right] \).

15.5.16 Two-dimensional random variables and their characteristics

4.99 $

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?

15.5.17 Two-dimensional random variables and their characteristics

6.24 $

Problem: The random variable \( \xi \) is distributed according to Bernoulli's principle with the parameter \( p=0,65 \). Make the distribution table of the random vector \( \eta=\left(\xi_{1}, \xi_{2}\right)^{T} \), where \( \xi_{1}=\xi \), and \( \xi_{2}=\xi^{3} \). 1. Calculate the expected value of the random vector \( E[\eta] \) and its covariance matrix \( V_{\eta} \). 2. Find all conditional distribution series for the random variable \( \xi_{1} \) under the condition that the random variable \( \xi_{2}=y_{j} \).

15.5.18 Two-dimensional random variables and their characteristics

3 $

Problem: Let's find the constant \( c \). Calculate the correlation coefficient \( \rho_{\xi_{1} \xi_{2}} \) and the probabilities of the following random events \( P\left\{\left(\xi_{1}=-2\right),\left(\xi_{2}<0\right)\right\}, P\left\{\xi_{2}>-1\right\} \), if the distribution table of a two-dimensional discrete random vector composed of these values has the form: \begin{tabular}{|c|c|c|c|} \hline\( y_{j} \) & -2 & -1 & 3 \\ \( x_{i} \) & & & \\ \hline-2 & 0.1 & 0.15 & \( c \) \\ \hline-1 & 0.15 & 0.25 & 0.15 \\ \hline \end{tabular}

15.5.19 Two-dimensional random variables and their characteristics

3 $

Problem: From an urn, containing 6 white and 4 black balls, Jon and Peter take out 3 balls, without returning them, in the following order: Peter - Jon - Jon. Random variables: \( X \) is the number of white balls, that Jon has, \( Y \) is the number of white balls, that Peter has. Describe the distribution law of the random vector \( (X, Y) \). Find the correlation coefficient \( p[X, Y] \).

15.5.20 Two-dimensional random variables and their characteristics

3 $

Problem: The system of random variables \( (X, Y) \) obeys the normal law with numerical characteristics \( M[X]=M[Y]=0, E_{x}=E_{y}=10, k_{x y}=0 \). Determine the probability that a) \( X0 ; Y<0 \).

15.5.21 Two-dimensional random variables and their characteristics

3.75 $

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