  In this paper, we construct an intermediate distribution linking the Gaussian
and the Cauchy distribution. We provide the probability density function and
the corresponding characteristic function of the intermediate distribution.
Because many kinds of distributions have no moment, we introduce weighted
moments. Specifically, we consider weighted moments under two types of weighted
functions: the cut-off function and the exponential function. Through these two
types of weighted functions, we can obtain weighted moments for almost all
distributions. We consider an application of the probability density function
of the intermediate distribution on the spectral line broadening in laser
theory. Moreover, we utilize the intermediate distribution to the problem of
the stock market return in quantitative finance.
