A multivariate Chebyshev bound of the selberg form

The least upper bound for the probability that a random vector with fixed mean and covariance will be outside the ball is found. This probability bound is determined by solving a scalar equation and, in the case of identity covariance matrix, is given by an analytical expression, which is a multivariate generalization of the Selberg bound. It is shown that at low probability levels, it is more typical when the bound is given by the new expression if compared with the case when it coincides with the right-hand side of the well-known Markov inequality. The obtained result is applied to solving the problem of hypothesis testing by using an alternative with uncertain distribution.