Generalized cauchy distributions

Statistical distributions of the form C(l + [' x | )~ are considered. For k = 2, m = 1 this is the Cauchy distribution. For k = 2, m == ^(n -\1), % == n~&, it is the Student-Fisher distribution with n degrees of freedom. The Cauchy distribution is of interest because the variance of the mean is infinite while that of the median is finite. In this paper a study is made of the case m = 1. The value of k is determined for which the variance of the mean and that of the median are equal. The distribution of the mean for the case k = 4, m = 1 is derived. Several other special cases are studied.