One dimensional random variables and their characteristics

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Problem: The batch contains 20 televisions, among which 6 have a defect. Two televisions have been bought. Compose the series of serviceable televisions among the bought ones. Find numerical characteristics of the given random variable.

15.2.1 One dimensional random variables and their characteristics

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Problem: Independent random variables \( x \) and \( y \) have the distribution \begin{tabular}{|c|c|c|} \hline\( x_{i} \) & -2 & -1 \\ \hline\( p_{i} \) & 0,4 & 0,6 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline\( y_{i} \) & 1 & 2 \\ \hline\( p_{i} \) & 0,7 & 0,3 \\ \hline \end{tabular} Find the law of distribution of the random variable \( Z=-x+2 y \). Find \( F(z), P(3 \leq z \leq 4) \).

15.2.2 One dimensional random variables and their characteristics

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Problem: The distribution density of the random variable \( \mathrm{X} \) is given: \[ f(x)=\left\{\begin{array}{cc} 0, & x \leq 2 \\ 2 x-4, & 23 \end{array}\right. \] Find \( F(x), M(x), D(x), \sigma(x) \).

15.2.3 One dimensional random variables and their characteristics

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Problem: The lifespan of an electric lamp has an exponential distribution with a mathematical expectation of \( L \) hours. What is the possibility that the lamp will last from \( m_{1} \) to \( M_{1} \) hours if \[ L=76 ; m_{1}=75 ; M_{1}=109 \text {. } \]

15.2.4 One dimensional random variables and their characteristics

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Problem: An experiment is performed in which a random occurrence \( A \) can take place with \( p \) probability. The experiment is repeated under the same conditions \( n \) times. \( n=1000 ; p=0,6 \). Determine the probability that the occurrence \( A \) will take place at least 580 times.

15.2.5 One dimensional random variables and their characteristics

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Problem: The random variable \( X \) in the interval \( (0 ; 1) \) is given by the distribution density \( f(x)=2 x \), outside this interval \( f(x)=0 \). Find the initial and central moments of the first, second, third and fourth orders.

15.2.6 One dimensional random variables and their characteristics

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Problem: The one-dimensional random variable \( \xi \) is given by the distribution density \( P(x)=\gamma e^{a x^{2}+b x+c} \), where \( a=-3, b=-4, c=0, x_{1}=\frac{1}{3}, \quad x_{2}=\frac{4}{3} \). Find the constant \( \gamma \), the expected value, the distribution function and the probability that the value \( \xi \) belongs to the integral \( \left[x_{1}, x_{2}\right] \).

15.2.7 One dimensional random variables and their characteristics

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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

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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: 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

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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

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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

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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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