15.2.58 One dimensional random variables and their characteristics
Condition: Random variable \( X \) has a distribution function \[ f(x)=\left\{\begin{array}{cl} 0, & \text { for } x<0 \\ \frac{4\left(a x-x^{2}\right)}{a^{2}}, & \text { for } x \in\left[0, \frac{a}{2}\right] \\ 1, & \text { for } x>\frac{a}{2} \end{array}\right. \] Find: \[ M(x), \quad P\left(x<\frac{a}{4}\right), \quad P\left(\frac{a}{8}
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.