An intermediate distribution between Gaussian and Cauchy distributions

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.