Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:, \tag{1}\end{equation} as a measure of distance of $f$ from $g$ at point $\beta$. Two questions:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. Is there any relationship between (1) and the well-known distance (or divergence) measures?

*Definition of statistical divergence: https://en.wikipedia.org/wiki/Divergence_(statistics)

*Definition of statistical distance: https://en.wikipedia.org/wiki/Statistical_distance