Suppose we are driving a car and we see a red light at a distance $d$. We don't know when it will turn green again but we know the period of red lights, that is, an upper limit for waiting. We want to minimize the expected loss of energy due to braking. What should be our slowing curve? (assuming we're driving at adequately high speeds)
If we knew the remaining time $t$ exactly, then the solution would simply be constant speed. We would adjust our speed as quickly as possible to the value $d/t$. (actually $d/t$ is possible only when we are able to decrease speed instantly.)
However the probabilistic version involves continuous Bayesian updates I think which I'm not really good at. Anyway, I follow some reasoning and end up with a parabolic displacement-time curve which ends up with zero speed at the worst case scenario(which we see the red light just after it has turned from green).
