Joint distributions and covariance
How strongly are the values of Brownian motion at two different times related?
Goal: measure that Cov(W(s), W(t)) = min(s, t), and see that the part independent of W(s) is exactly the increment.
Drag the shaded window to slide it, its edges to resize. The coloured paths reappear as same-coloured points on the joint chart.
W(s) = 0.80 ⇒ W(t) ~ N(0.80, 0.40). The dashed line is E[W(t) | W(s)] = W(s): the best guess is where you are now. Wherever you drag the slice, the conditional sd stays √(t − s).