  The Weibull distribution can be obtained using a power transformation from
the standard exponential distribution. In this article, we will consider a
symmetrized power transformation of a random variable with the standard normal
distribution. We will call its distribution the $\alpha$-{\it normal (Gaussian)
distribution}. We examine properties of this distribution in detail. We
calculate moments and consider the moment problem of $\alpha$-normal
distribution. We derive the formula of its differential entropy and
(exponential) Orlicz norm. % of $\alpha$-normal random variables. Moreover, we
define the joint distribution function of the multivariate $\alpha$-normal
distribution as a meta-Gaussian distribution with $\alpha$-normal marginals. We
consider also the limiting distribution as $\alpha$ tends to infinity.
