15.2.61 One dimensional random variables and their characteristics
Condition: Random variables \( \quad X_{k}, k=1,2, \ldots, n \), distributed uniformly on the segment \( [0, a] \). Find the distribution function and probability density function of the random variable \( Z=\min _{1 \leq k \leq n}\left\{X_{1}, X_{2}, \ldots, X_{n}\right\} \), counting the values \( X_{k} \) independent. Calculate the expectation and variance of the random variable \(Z\). Calculate the probability \( P(Z \leq a / 2) . a=10, n=3 \).
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.