I have a question of calculating bias in variance shadow mapping.

I saw a article of vsm, in 8.4.2 Biasing Section, that explain how to avoid self-shadowing using biasing equation using in vsm.

expanding equation of biasing line:2

expanding equation of biasing line:3

(I’m sorry to not post image, because of my reputation is lower than 10. please see and compare a article that I link at first)

During expanding of biasing equation, writer of article uses gaussian distribution with a half-pixel standard deviation(1/2) to calculate E(x^2) and E(y^2). Why it can do?

expanding equation of biasing line:1

I think f is surface of depth. and its x, y argument is like offset. if that is origin, f returns mean of depth. With assuming x, y is uniform distribution, E(x) = E(y) = 0 can be thinking.

But I have no idea why use 1/2 as x’s, y’s standard deviation.
Is just assuming that 68% of x or y is in pixel by 68-95-99.7 rule and is in order to calculate E(x^2) by restricting E(x) = E(y) = 0?