15.2.10 One dimensional random variables and their characteristics

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 \( (\alpha, \beta) \). \[ f(x)=\left\{\begin{array}{lc} 0, & \text { when } x<1, \\ A x^{3}, & \text { when } 1 \leq x \leq 2, \quad \alpha=1,1, \quad \beta=1,5 . \\ 0, & \text { when } x>2 \end{array}\right. \]