Title: 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.