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Problem: The random variable \( \xi \) is given by the density of the distribution \( P_{\xi}(x) \). The random variable \( \eta \) is the area of a regular triangle with side \( \xi \). For the random variable \( \eta \) find the distribution function, the density of the distribution, the expected value and the dispersion, where \[ \begin{array}{l} P_{\xi}(x)=\left\{\begin{array}{l} \frac{2(x-a)}{(b-a)^{m}}, \quad x \in[a, b], \\ 0, \quad x \notin[a, b] \end{array}\right. \\ a=2, \quad b=4, \quad m=2 . \end{array} \]

15.2.8 One dimensional random variables and their characteristics

2.61 $

Problem: The density of the probability distribution \( f(x) \) of the continuous random variable \( X \) is given. It is required: 1) determine the coefficient \( A \); 2) find the distribution function \( F(x) \); 3) plot the graphs of \( F(x) \) and \( f(x) \) schematically; 4) find the expected value and the dispersion of \( X \); 5) find the probability that \( X \) will take a value from the interval \( (\alpha, \beta) \). \[ f(x)=\left\{\begin{array}{lc} 0, & \text { when } x<1, \\ A x^{3}, & \text { when } 1 \leq x \leq 2, \quad \alpha=1,1, \quad \beta=1,5 . \\ 0, & \text { when } x>2 \end{array}\right. \]

15.2.10 One dimensional random variables and their characteristics

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Problem: Consider a random sample \( X_{i} \) from some known distribution and answer the following questions: a) find the estimate of the parameter \( A \) using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-1 ; A) \); b) find the estimate of the parameter \( B \), using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-B ; B) \); c) find the estimates of parameters \( c \) and \( C \) using the method of maximum likelihood estimation, if it is known that the sample is made from the uniform estimation \( U(c ; C) \); d) find (and compare) the estimates of the parameter \( L \) using the method of moments and the method of maximum likelihood estimation, if it is known that the sample is made from the exponential distribution \( E_{L} \); e) find the estimate of the parameter \( m \) using the method of moments, if it is known that the sample is made from the normal distribution \( N(m, 1) \); f) find the estimates of the parameters \( M \) and \( S \) using any known method, if it is known that the sample is made from the normal distribution \( N(M, S) \); g) plot a histogram and a polygon based on the sample, the number of intervals is 3 ; h) in each of the points (a) - (f) estimate the proximity of this theoretical distribution to the empirical one

20.7 Mathematical statistics

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3 Problem: Calculate the volume of the solid: \[ V:\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}} \leq \ln \frac{x^{2}+y^{2}+z^{2}}{x^{2}+y^{2}} . \]

9.5.8 Volume of a solid

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30 Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} y=\sqrt{x^{2}+z^{2}} \\ z=2-x^{2}-z^{2} \end{array} .\right. \]

9.5.13 Volume of a solid

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(a) Problem: Calculate the volume of the solid bounded by the given surfaces using a triple integral. Plot the drawings of the given solid and its projection on plane \( X O Y \). \[ x^{2}+y^{2}=2 y, \quad z=\frac{13}{4}-x^{2}, \quad z=0 \]

9.5.17 Volume of a solid

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a) Problem: Find the volume of a cylindrical solid bounded by surfaces \( z=16-y^{2}, x^{2}+y^{2}=2 x, z=0 \).

9.5.18 Volume of a solid

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(3) Problem: Calculate the volume of a solid bounded by the given surfaces using a triple integral. Plot drawings of this solid and its projection on plane XOY. \[ z=0, \quad z-4 \sqrt{y}=0, \quad x=0, \quad x+y=4 \]

9.5.19 Volume of a solid

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(a) Problem: Calculate the volume of the solid bounded by the surfaces: \[ z=\sqrt{x^{2}+y^{2}}, \quad z=1-\sqrt{1-x^{2}-y^{2}} . \]

9.5.14 Volume of a solid

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3) Problem: Calculate the volume of a solid bounded by the given surfaces using a triple integral. Plot the drawings of the given solid and its projection on XOY. \[ z=0, \quad z=4-x-y, \quad x^{2}+y^{2}=4 \]

9.5.15 Volume of a solid

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Problem: Calculate the mass of the plate \( D \) with density, where \[ \mu=\frac{2 y-3 x}{x^{2}+y^{2}}, \quad D:\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ x^{2}+y^{2}=16 \\ x=0, y=0,(x \leq 0, y \geq 0) \end{array}\right. \]

9.1.4 Double integrals

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Problem: Calculate the mass of the plate \( D \) with the density, where \[ D:\left\{\begin{array}{l} \frac{x^{2}}{16}+y^{2} \leq 1 \\ x \geq 0, y \geq 0 \end{array} \quad \mu(x, y)=5 x y^{7} .\right. \]

9.1.5 Double integrals

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Problem: Find the double integral over the area \[ D:\left\{\begin{array}{l} x=1, y=x^{3} \\ y=-\sqrt[3]{x} \end{array}\right. \text {. } \]

9.1.6 Double integrals

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Problem: Change the order of integration in the iterated integral: \[ \int_{0}^{1} d x \int_{0}^{x} f d y+\int_{1}^{\sqrt{2}} d x \int_{0}^{\sqrt{2-x^{2}}} f d y . \]

9.1.7 Double integrals

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Problem: Find the statistical moment about the axis \( O X \) of the plate \( D \) with density, where \[ D:\left\{\begin{array}{c} x=\sqrt{4-y^{2}} \\ x=0 \end{array}\right. \]

9.1.8 Double integrals

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