How close are two Gaussian random variables? Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
 A: As the measure of the closeness of two distributions $p_A$ and $p_B$ You could use the Bhattacharyya coefficient
$$w=\int \sqrt{p_A(x)p_B(x)}\,dx\in[0,1],$$
which for two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$) is given by $w=e^{-d}$ with
$$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}. $$
A: This depends really on the use you will have of that distance ... For many (statistical) purposes an asymmetric divergence is more natural than a symmetric distance. So look into the Kullback-Leibler divergence, for univariate normals see https://stats.stackexchange.com/a/7449/11887, for the multivariate case see https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence#Multivariate_normal_distributions.
For help with interpretations see https://stats.stackexchange.com/questions/188903/intuition-on-the-kullback-leibler-kl-divergence/189758#189758
A: The Kolmogorov distance is a possible answer. I derived it in this article of my former blog (at the end of the article).
