On an example of expectation evaluation

We study the distribution of the maximal element $\overline{\xi}_n$ of a sequence of (possibly) independent random variables $\xi_1,\dots,\xi_n$. A formula for evaluation of a random variable expectation based on a quantile function is considered. This formula is applied to evaluation of the expectation for a nondecreasing function of a random variable transformed via its distribution function. The case of a discontinuous distribution function is the most interesting. As a corollary, we refine an example proposed in the author's previous article [Theory Probab. Appl., 68 (2023), pp. 392–410].