# What is the name of $D(F,G)=\int(F(x)-G(x))^2dF(x)$?

Is there a statistical distance between two 1-dim distribution F and G that $$D(F,G)=\int(F(x)-G(x))^2dF(x)$$?

Or to symmetrize it, take $$D^s(F,G)=\int(F(x)-G(x))^2dF(x)+dG(x)$$

If not, why? (What are the main disadvantages?)

I learned that $$D_E(F,G)=\int(F(x)-G(x))^2dx$$ is famous. But what about $$D(F,G)=\int(F(x)-G(x))^2w(x)dx$$? I want to check relevant materials for learning, but don't known how to start.

• And do you know if there is a distance with form like $\int (F(x)-G(x))^2w(x)dx$ or $\int (F(x)-G(x))^2w(x)dF(x)$ – Happy Superman Mar 27 at 3:52