  We provide an exact expressions for the 1-Wasserstein distance between
independent location-scale distributions. The expressions are represented using
location and scale parameters and special functions such as the standard
Gaussian CDF or the Gamma function. Specifically, we find that the
1-Wasserstein distance between independent univariate location-scale
distributions is equivalent to the mean of a folded distribution within the
same family whose underlying location and scale are equal to the difference of
the locations and scales of the original distributions. A new linear upper
bound on the 1-Wasserstein distance is presented and the asymptotic bounds of
the 1-Wasserstein distance are detailed in the Gaussian case. The effect of
differential privacy using the Laplace and Gaussian mechanisms on the
1-Wasserstein distance is studied using the closed-form expressions and bounds.
