Let's say that I pick two random $x$ values on $[-1,1]$ and square each value to get the correspond $y$ value, thus: $$(x_1, x_1^2), (x_2, x_2^2).$$ I then find the equation for the line connecting these two points.
Say I repeated this $1,000$ times (or $10,000$, or $100,000$, etc.). How could I determine the average equation that I would get?
For any randomly-chosen $x_1$ and $x_2$, the line passing through $(x_1, x_1^2)$ and $(x_2, x_2^2)$ is $$y = (x_1 + x_2) x -x_1 x_2.$$ By taking expectations, we get $$E[y] = y=E[x_1+x_2]x-E[x_1 x_2].$$ Since $x_1$ and $x_2$ are chosen independently, and $E[x_1]=E[x_2]=0$, we have $E[x_1 x_2]=E[x_1]E[x_2]=0$. Consequently, the equation of the line you are looking for is $$y = 0.$$