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?



How close are two Gaussian random variables?
As the measure of the closeness of two distributions $p\_A$ and $p\_B$ You could use the [Bhattacharyya coefficient](https://en.wikipedia.org/wiki/Bhattacharyya_distance)
$$w=\int \sqrt{p\_A(x)p\_B(x)}\,dx\in[0,1],$$
which for two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$) is given by $w=e^{-d}$ with
$$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}. $$


