  The q-Gaussian is a probability distribution generalizing the Gaussian one.
In spite of a q-normal distribution is popular, there is a problem when
calculating an expectation value with a corresponding normalized distribution
and not a q-normal distribution itself.
  In this paper, two q-moments types called normalized and unormalized
q-moments are introduced in details. Some properties of q-moments are given,
and several relationships between them are established, and some results
related to q-moments are also obtained. Moreover, we show that these new
q-moments may be regarded as a generelazation of the classical case for q = 1.
Firstly, we determine the q-moments of q-Gaussian distribution. Especially, we
give explicitely the kurtosis parameters. Secondly, we compute the expression
of the q-Laplace transform of the q-Gaussian distribution. Finally, we study
the distribution of sum of q-independent Gaussian distributions.
