The geometrical origins of some distributions and the complete concentration of measure phenomenon for mean-values of functionals

Abstract: We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form functionals in the balls of continuous functions space with $p-$norm, and show the complete concentration of measure phenomenon which means that a functional takes its average on a ball with probability 1, from which we have nonlinear exchange formula of expectation.