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Problem: Find the distribution function of the random variable $ \tau=[\xi] $, if the random variable $ \xi $ is distributed in accordance with the exponential law of the distribution with the parameter $ \lambda=5 $. The notation $ [\xi] $ stands for the integer of the number.

15.2.18 One dimensional random variables and their characteristics

1 $

Problem: Let the random variable $ \xi $ be the sum of the amount of the loan, requested by the bank's clients per day. This amount can change in the range from 0 to 1000000 (USD). The bank immediately withdraws the commissions of $ 1 \% $. What function will describe the amount of commissions, received by the bank, depending on the requested loan amount (the random variable $ \tau $ )? Find the distribution function of the random variable $ F_{\tau}(z) $.

15.2.19 One dimensional random variables and their characteristics

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Problem: During the observed period the changes in the exchange rate of EUR in relation to the USD can be considered a random variable, with equal probability taking values from the interval $ [28.3 ; 30] $. The amount of tuition fees at the institute is calculated by the following rate, linked by the exchange rate of the following function $ \tau=\xi-1,7 $. Find the distribution density and function of the random variable $ \tau $. Calculate its expected value and dispersion.

15.2.20 One dimensional random variables and their characteristics

1.75 $

Problem: When calculating the compound interests, the calculation of the real interest rate in terms of inflation is carried by the formula: \[ \tau=\frac{1+r}{1+\xi}-1 . \] The given (announced) gross rate is denoted by $ r $, and the rate of price growth for a year is denoted by $ \xi $. Assuming that $ \xi $ is a random variable, equally distributed on $ [10 \% ; 12 \%] $, find the distribution function of the real interest rate.

15.2.21 One dimensional random variables and their characteristics

1.5 $

Problem: The amount of money that a bank depositor, who has put 1000 dollars into his account, will get in three years, is calculated by the formula: \[ \tau=1000(1+\xi)^{3} . \] The annual interest rate, which, according to the conditions of the agreement, can be changed by the bank unilaterally, is denoted by $ \xi $. Assuming that the random variable $ \xi $ is equally distributed on the segment $ [0.5 \% ; 2 \%] $, Find the distribution law in the form of density and the distribution function of the random variable $ \tau $. Calculate the expected value and the dispersion $ E[\tau], V[\tau] $.

15.2.22 One dimensional random variables and their characteristics

2.5 $

Problem: There is a safe in the depository of the bank, where $ 80.000 \$ $ are kept. During the day, some amount of money (random variable $ \xi $ ), not known in advance, can be demanded from the safe. The distribution table of the random variable $ \xi $ has the following form: \begin{tabular}{|c|c|c|c|c|c|} \hline$ x_{i} $ & $ 10000 \$ $ & $ 15000 \$ $ & $ 20000 \$ $ & $ 25000 \$ $ & $ 30000 \$ $ \\ \hline$ p_{i} $ & 0.2 & 0.15 & 0.25 & 0.19 & 0.21 \\ \hline \end{tabular} 1. Make the distribution table of the random variable $ \tau $-the amount of money, remaining in the safe, if $ \tau=80000-\xi $. 2. Find the distribution function of this random variable $ \tau $. 3. Calculate the expected value and the dispersion $ E[\tau], V[\tau] $.

15.2.23 One dimensional random variables and their characteristics

2.5 $

Problem: The distribution table of the discrete random variable $ \xi $ has the form: \begin{tabular}{|c|c|c|c|c|c|} \hline$ x_{i} $ & $ -\pi / 2 $ & $ -\pi / 6 $ & $ \pi / 2 $ & $ \pi / 6 $ & $ \pi $ \\ \hline$ p_{i} $ & 0.3 & 0.2 & 0.1 & 0.1 & 0.3 \\ \hline \end{tabular} 1. Make the distribution tables and find the distribution functions for the random variables $ \tau_{i} $, $ i=1,2,3 $, if: a) $ \tau_{1}=\cos \xi $ b) $ \tau_{2}=\sin \xi $ c) $ \tau_{3}=|\sin \xi| $. 2. Calculate the expected values $ E\left[\tau_{i}\right] $ and the dispersions $ V\left[\tau_{i}\right], i=1,2,3 $.

15.2.24 One dimensional random variables and their characteristics

4.99 $

Problem: When a dice is rolled, it's equally probable that numbers 1-6 appear on the top. The random variable $ \tau $ is equal to the sum of the numbers appearing as a result of two independent rolls. Make the distribution table of the random variable $ \tau $.

15.2.25 One dimensional random variables and their characteristics

1.75 $

Problem: The continuous random variable $ \xi $ is equally distributed on the segment $ [-5,1] $. It's known that: a) the random variable $ \tau_{1}=4-\xi $, b) the random variable $ \tau_{2}=|\xi| $. Find: 1. The distribution density of $ f_{\tau}(z) $ for each random variable $ \tau_{1} $ and $ \tau_{2} $. 2. Calculate the expected value and the dispersion $ E\left[\tau_{i}\right], V\left[\tau_{i}\right], i=1,2 $.

15.2.26 One dimensional random variables and their characteristics

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Problem: Random variables $ \tau $ and $ \xi $ are connected to each other functionally: $ \tau=\arctan \xi $. It's known that the random variable $ \xi $ is continuous, with the next distribution density: \[ f_{\xi}(x)=\left\{\begin{array}{cc} 0, & \text { if } x<2 \\ c / x^{5}, & \text { if } x \geq 2 \end{array}\right. \] 1. Find the constant $ c $. 2. If we can claim that the random variable $ \tau $ will be continuous, find the expression for the density of $ f_{\tau}(z) $. 3. Write the expression for the distribution function of the random variable $ \tau $. 4. Calculate $ E[\tau], V[\tau] $ and the probability of $ p\{2<\tau<10\} $.

15.2.27 One dimensional random variables and their characteristics

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Problem: It's known that the continuous random variable $ \xi $ is distributed in accordance with the normal law of distribution with the parameters $ m=-1 $ and $ \sigma=4 $. 1. Find the laws of distribution in the form of distribution density and function of the following random variables: a) $ \tau_{1}=\sqrt[3]{\xi} $ b) $ \tau_{2}=-\xi^{2} $ c) $ \tau_{3}=0.5 \xi-1 $. 2. Calculate the expected values $ E\left[\tau_{3}\right], E\left[\tau_{2}\right] $.

15.2.28 One dimensional random variables and their characteristics

3.75 $

Problem: A demand deposit account was opened in the amount of 20 thousand dollars at a simple annual interest rate of $ 1 \% $. While closing the account the depositor will get the sum: $ \tau=20000(1+0.01 \xi) $ Experience shows that the time after which the depositor can close the account on such a deposit (random variable $ \xi $ ), can be approximated by an exponential distribution law with the parameter $ \lambda=2 $. 1. What is the average time spent on such a deposit? 2. What is the distribution density of the random variable $ \tau $ ? 3. What is the average variable of the obtained amount of money?

15.2.29 One dimensional random variables and their characteristics

2.5 $

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

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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

2.5 $

Problem: The ball is tossed into the basketball hoop. The probability of a success by one toss is equal to 0,7 . Let the random variable $ \xi_{1} $ be the number of successful tosses, and the random variable $ \xi_{2} $ be the number of misses by one toss. 1. Make a table of the joint distribution of these random variables. 2. Calculate the expected values of these random variables and write down the expected value of the vector $ \eta=\left(\xi_{1}, \xi_{2}\right)^{T} $. 3. Calculate the dispersions, the correlation moment and the correlation coefficient of these random variables. Write the covariance and correlation matrices. 4. Will these random variables be connected to each other linearly?

15.5.6 Two-dimensional random variables and their characteristics

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