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Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=9 u_{x x}, \quad u(x, 0)=9 \sin 3 \pi x-5+2 x, \\ u(0, t)=-5, \quad u(3, t)=1 . \end{array} \]

11.5.5.13 Mixed problems

6.24 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{16} u_{x x}+10 \sin 3 t \cdot \sin 4 x, \quad u(x, 0)=0, \\ u(0, t)=0, \quad u(\pi, t)=0 . \end{array} \]

11.5.5.14 Mixed problems

6.24 $

Problem: Solve the external Dirichlet boundary-value problem for the circle: \[ \begin{array}{l} \Delta u(r, \varphi)=0, \quad r>r_{0}, \quad 0 \leq \varphi<2 \pi, \\ |u(r, \varphi)|<+\infty, \quad u\left(r_{0}, \varphi\right)=\frac{u_{0} \sin \varphi}{25+7 \cos \varphi} . \end{array} \]

11.5.5.15 Mixed problems

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Problem: Determine the type of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} u_{x}+v_{x}-u_{y}+v_{y}=x y \\ 2 v_{y}-v_{x}+u_{x}+3 u_{y}=u \end{array}\right. \]

11.6.1 Systems of equations in partial derivatives of the first order

2.5 $

Problem: Determine the type of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} u_{y}+3 v_{x}-v_{y}=y \\ 2 v_{y}-v_{x}=x \end{array}\right. \]

11.6.2 Systems of equations in partial derivatives of the first order

2.5 $

Problem: Determine the type of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} 3 u_{x}+4 u_{y}-v_{y}+v_{x}=0 \\ u_{x}+v_{y}-v_{x}=0 \end{array}\right. \]

11.6.3 Systems of equations in partial derivatives of the first order

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Problem: The density of the distribution of the random variable \( X \) has the form \[ f(x)=\left\{\begin{array}{l} 0, \quad \text { when } x<0 \\ \frac{x^{m}}{m !} e^{-x}, \text { when } x>0 \end{array}\right. \] Find \( M X \) and \( D X \).

15.2.9 One dimensional random variables and their characteristics

1.75 $

Problem: The distribution density of the random variable \( \xi \) is given: \[ f_{\xi}(x)=\left\{\begin{array}{c} 0, \text { if } x \leq-4, \\ \frac{c}{16} x+\frac{1}{4}, \text { if }-44 . \end{array}\right. \] The random variable \( \tau \) is equal to: \[ \tau=\left\{\begin{array}{cc} \xi, & \text { if } \xi<0, \\ -\xi, & \text { if } \xi \geq 0 . \end{array}\right. \] Find 1. the constant \( c \). 2. the distribution density and the expected value of the random variable \( \tau \).

15.2.30 One dimensional random variables and their characteristics

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Problem: The distribution table of the discrete random variable \( \xi \) has the form: \begin{tabular}{|c|c|c|c|c|c|} \hline\( x_{\mathrm{i}} \) & -2 & -1 & 0 & 1 & 2 \\ \hline\( p_{i} \) & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ \hline \end{tabular} Make distribution tables for random variables \( \tau_{i}, i=1,2,3 \) if: \( \tau_{1}=-\xi, \tau_{2}=|\xi|, \tau_{3}=\xi^{2} \). Determine the expected values \( E\left[\tau_{i}\right] \) and the dispersions \( V\left[\tau_{i}\right], i=1,2,3 \).

15.2.11 One dimensional random variables and their characteristics

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Problem: When tossing a 4-sided pyramid, it is equally likely to get from 1 to 4 numbers. The random variable \( \tau \) is equal to the sum of the numbers, from two independent tosses. Make the distribution table of this random variable \( \tau \).

15.2.12 One dimensional random variables and their characteristics

2.5 $

Problem: The exchange rate of the US dollar against Euro can be described by the following function: \( \tau=\alpha \xi \), where \( \alpha \) is a parameter. The change in the Euro exchange rate is given by the following distribution table: \begin{tabular}{|c|c|c|c|c|c|c|} \hline\( x_{i} \) & 11,7 & 12 & 12,3 & 12,5 & 13 & 13,1 \\ \hline\( p_{i} \) & 0,1 & 0,1 & 0,2 & 0,15 & 0,2 & 0,25 \\ \hline \end{tabular} What will the distribution table of the random variable \( \tau \) be like? Find the expected value and the dispersion of \( \tau \).

15.2.13 One dimensional random variables and their characteristics

1.25 $

Problem: Random variables \( \tau \) and \( \xi \) are functionally related to each other: \( \tau=1 / \xi \). It's known that the random variable \( \xi \) is continuous and has the following distribution density: \[ f_{\xi}(x)=\left\{\begin{array}{ll} 0, & \text { if } x<1 \\ 3 / x^{4}, & \text { if } x \geq 1 \end{array}\right. \] Is it possible to claim that the random variable \( \tau \) will also be continuous? Justify the answer. Find the expression of the density \( f_{\tau}(z) \). Calculate the expected value, the dispersion \( E[\tau], V[\tau] \) and the possibility \( p\{0,1<\tau<0,3\} \).

15.2.14 One dimensional random variables and their characteristics

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Problem: It's known that the continuous random variable \( \xi \) is distributed according to the exponential distribution law with the parameter \( \lambda=0,2 \). Find the laws of distribution in the form of a density and distribution function of the following random variables: \[ \begin{array}{l} \text { 1. } \tau_{1}=\sqrt{\xi} \\ \text { 2. } \tau_{2}=\xi^{2} ; \\ \text { 3. } \tau_{3}=0,5 \ln (\xi) . \end{array} \] Calculate their numerical characteristics.

15.2.15 One dimensional random variables and their characteristics

6.24 $

Problem: The figure shows graphs of functions relating two random variables \( \tau \) and \( \xi \); \( \tau=g(\xi) \). It's known that \( \xi \) is a continuous random variable. In which case can we claim that the random function \( \tau \) will also be continuous? Justify the answer.

15.2.16 One dimensional random variables and their characteristics

0.75 $

Problem: The continuous random variable \( \xi \) is evenly distributed on the segment \( [-2,2] \), and the random variable \( \tau=\xi^{2} \). Find \( f_{\tau}(z), E[\tau] \) and \( V[\tau] \). Solve this problem, assuming that the random variable \( \xi \) is evenly distributed on the segment \( [-2,3] \).

15.2.17 One dimensional random variables and their characteristics

3 $

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