15.2.57 One dimensional random variables and their characteristics
\( \underline{\mathrm{y}_{\text {word: }}} \) The random variable \( X \) has a distribution function \[ f(x)=\left\{\begin{aligned} 0, & \text { at } x<0 \\ \sqrt{x-a}, & \text { for } x \in[a, a+1] \\ 1, & \text { for } x>a+1 \end{aligned}\right. \] Find: \[ \begin{array}{ll} M(X), \quad D(X), & P\left(X\frac{1}{4}\right), & a=19 \end{array} \]
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.