15.2.62 One dimensional random variables and their characteristics
Condition: The random variable is specified by the distribution function \( F(x) \). Find the probability distribution density \( f(x) \), the mathematical expectation and variance of the random variable, the probability of falling into the interval \( (\alpha ; \beta) \). Construct graphs of functions \( F(x), f(x) \). \[ F(x)=\left\{\begin{array}{cc} 0, & x \leq 1 \\ \frac{x^{2}-x}{6}, & 13 \end{array} \quad(0 ; 1)\right. \]
Study of discrete and continuous one-dimensional random variables, calculation of their characteristics such as mathematical expectation, variance, standard deviation, moments, distribution and density functions. We also consider problems on known distributions - Gauss, Bernoulli, Poisson. Finding the probabilities of various events, including those from everyday life.