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Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ (1+\sqrt{z-x-y}) \cdot \frac{\partial z}{\partial x}+x \frac{\partial z}{\partial y}=2, \quad z=x \text { when } y=0 . \]

11.4.41 First order partial differential equations

3.85 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}-y \frac{\partial z}{\partial y}=z^{2}(x-3 y), x=1, \quad y z+1=0 . \]

11.4.42 First order partial differential equations

3.34 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ (y-z) \frac{\partial z}{\partial x}+(z-x) \frac{\partial z}{\partial y}=x-y, \quad z=y=-x . \]

11.4.43 First order partial differential equations

3.85 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=z-x y, \quad x=2, \quad z=y^{2}+1 . \]

11.4.44 First order partial differential equations

3.85 $

Problem: The probability that the element has passed the QC is equal to 0,8. Estimate by Chebyshev's inequality the probability that among 400 randomly chosen elements there will be from 60 to 100 unverified elements. Specify the result with the help of Laplace integral theorem.

15.2.31 One dimensional random variables and their characteristics

3.08 $

Problem: Random errors in measuring the distance to the fixed target are subject to the normal law of distribution with the expected value of \( a=5 \mathrm{~m} \) and the dispersion \( \sigma^{2}=100 \mathrm{~m}^{2} \). Determine the probability that: a) the measured value of the distance deviates from the true value by more than \( 15 \mathrm{~m} \). b) for three independent measurement the error of at least one measurement will not exceed \( 15 \mathrm{~m} \) in the absolute value.

15.2.32 One dimensional random variables and their characteristics

2.57 $

Problem: The random variable \( X \) a distribution, the probability density of which is \( f(X)=\frac{\lambda}{2} e^{-\lambda|x|}(\lambda>0) \). Find the expected value and the dispersion of the random variable \( Z=2+0,5 X \).

15.2.33 One dimensional random variables and their characteristics

2.06 $

Problem: Let \( \xi_{1}, \cdots, \xi_{n} \) are independent identically distributed random variables, with finite dispersion \( \sigma^{2} \). Let's designate \[ \bar{\xi}=\frac{\xi_{1}+\cdots+\xi_{n}}{n} . \] Find the expected value of the random variable \[ \frac{1}{n-1} \sum_{k=1}^{n}\left(\xi_{k}-\bar{\xi}\right)^{2} . \]

15.2.34 One dimensional random variables and their characteristics

6.42 $

Problem: Let \( \xi \) is a random variable with a symmetrical distribution. Let's substitute \[ \eta=\left\{\begin{array}{l} \xi, \text { when }|\xi| \leq c \\ 0, \text { when }|\xi|>c \end{array}, c>0 .\right. \] Let's denote \( f(t) \) and \( g(t) \) are the characteristic functions of \( \xi \) and \( \eta \) respectively. Prove that there will be such \( \varepsilon>0 \), that \( f(t) \leq g(t) \), when \( |t| \leq \varepsilon \).

15.2.35 One dimensional random variables and their characteristics

3.85 $

Problem: It's known that for independent random variables \( X_{1}, \cdots, X_{4} \) their expected values are \( E\left(X_{i}\right)=-2 \), the dispersions are \( D\left(X_{i}\right)=1 \), \( i=1, \cdots, 4 \). Find the dispersion of the product \( D\left(X_{1} ; \cdots X_{4}\right) \).

15.2.36 One dimensional random variables and their characteristics

0 $

Problem: The probability of a stock rising by \( 3 \% \) in one working day is equal to 0,1 , the probability of rising by \( 0,1 \% \) is equal to 0,6 , and the probability of decrease by \( 1 \% \) is equal to 0,3 . Find the expected value of the change of the stock in 100 working days, taking into account that the initial stock is 1000 dollars, and the relative price changes for different working days are independent random variables.

15.2.37 One dimensional random variables and their characteristics

1.8 $

Problem: The random variable \( X \) is uniformly distributed on the segment \( [-3,11] \). Find the probability \[ P\left(\frac{1}{X-3}>4\right) \text {. } \]

15.2.38 One dimensional random variables and their characteristics

1.03 $

Problem: The random variable \( X \) is uniformly distributed on the segment \( [-4,2] \). Find \( E\left(e^{4 X}\right) \).

15.2.39 One dimensional random variables and their characteristics

0 $

Problem: The random variable \( X \) is distributed in accordance with the exponential law. Find the expected value \( E\left\{(X+8)^{2}\right\} \), if the dispersion \( D(X)=36 \).

15.2.40 One dimensional random variables and their characteristics

1.03 $

Problem: The student answers three questions on the examination card. The probability of a correct answer to the first question is equal to 0,5 , to the second one is 0,7 and to the third question 0,9 . It is required: a) make a distribution series of the number \( X \) of the questions, which are given correct answers; b) find the expected value and the dispersion of the random variable \( X \); c) accurately plot the graph of its function of the distribution \( F_{X}(x) \).

15.2.41 One dimensional random variables and their characteristics

3.08 $

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