One dimensional random variables and their characteristics

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Problem: Calculate the expected value and the dispersion of the hypergeometric distribution.

15.2.46 One dimensional random variables and their characteristics

3.82 $

Problem: The distribution function of the random variable is equal to 0 when \( x<-1, x^{2}+2 x+1 \) when \( -1 \leq x<0 \), and 1 for other \( x \). Find the expected value of this random variable.

15.2.47 One dimensional random variables and their characteristics

0.76 $

Problem: The distribution density of the random variable \( X \) is given. \[ f(x)=\left\{\begin{array}{cl} 0, & \text { when } x<0 \\ \alpha\left(x-\frac{x^{2}}{3}\right), & \text { when } 0 \leq x \leq 3 \\ 0, & \text { when } x>3 \end{array}\right. \] Find a) the coefficient \( \alpha \), b) the distribution function \( F(x) \), c) the expected value \( E(X) \) and the dispersion \( D(X) \).

15.2.48 One dimensional random variables and their characteristics

2.54 $

Problem: The random variables \( X_{1} \) and \( X_{2} \) have binomial and Poisson distributions, respectively. Find the probabilities of \( p\left(3 \leq X_{i} \leq 5\right) \), if the expected value is \( M\left(X_{i}\right)=3 \), and the dispersion is \( D\left(X_{1}\right)=1,875 \).

15.2.49 One dimensional random variables and their characteristics

2.04 $

Problem: The random variables \( X_{3} \) and \( X_{4} \) have an equal and normal distributions respectively. Find the probabilities of \( p\left(3

15.2.50 One dimensional random variables and their characteristics

2.54 $

Problem: The continuous random variable \( X \) has the distribution function \( F(x)=0 \) when \( \quad x<1, \quad F(x)=1 \quad \) when \( x>4 \), \( F(x)=c_{1}+(x / 4)+c_{2}(x-1)^{2} \) when \( 1

15.2.51 One dimensional random variables and their characteristics

2.04 $

Problem: Let the random variables \( X \) and \( Y \) be independent and given in accordance with the laws of distribution: \begin{tabular}{|c|c|c|c|} \hline\( x_{i} \) & 2 & 3 & 4 \\ \hline\( p_{i} \) & 0,3 & 0,2 & 0,5 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\( y_{i} \) & 0 & 1 & 2 & 3 \\ \hline\( q_{i} \) & 0,2 & 0,4 & 0,1 & 0,3 \\ \hline \end{tabular} Find the distribution law of the variable \( Z=X-Y \) and the expected value and dispersion of the variables \( X, Y \) and \( Z \).

15.2.52 One dimensional random variables and their characteristics

2.04 $

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