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?
gaussianpr.probabilityprobability distributions
Best Answer
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 $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}. $$