Suppose it is 1800 and you have just been employed as a captain by a shipping company that regularly transports goods between London and New York. The distance between London and New York is 3,000 nautical miles. Your employer tells you that on average the trip between London and New York takes 10 weeks so the average velocity of his ships is 300nm/week. However, because the ships are powered by sail (the steam ship has yet to be invented) there is a great of variance in the velocities of his ships. In fact, your data-minded employer tells you that the velocities of his ships for trips between London and New York are normally distributed with an average velocity of 300nm/week with a standard deviation of 50nm/week.
On your first voyage, you and I agree to pass the time by estimating every week what the final velocity of our trip will be. Every week we know how many total weeks we’ve been traveling and how many nautical miles of the original 3,000 remain between us and New York. We can thus calculate what our current velocity is.For instance, after two weeks 2,500nm remain between us and New York so our current velocity is:
(3,000nm-2,500nm)/2weeks = 250nm/week.
How can we best estimate what our final velocity for the entire trip will be using only the prior distribution of average velocities (mean=300nm/week, standard deviation=50nm/week), the distance remaining, and the number of weeks traveled?
How can we estimate the variance of our final velocity for the entire trip?
Keep in mind that we will want to update our estimate every week based on the new information of how far we’ve traveled.