A better model for the sailboat race of Problem 5.5.4 accounts for the fact that all boats are subject to the same randomness of wind and tide. Suppose in the race of ten sailboats, the finishing times Xi are identical Gaussian random variables with expected value 35 minutes and standard deviation 5 minutes. However, for every pair of boats i and j, the finish times Xi and Xj have correlation coefficient ρ = 0.8.
(a) What is the covariance matrix of X = [X1 ··· X10]?
(b) Let
Denote the average finish time. What are the expected value and variance of Y? What is P[Y ≤ 25]?
Problem 5.5.4
In a race of 10 sailboats, the finishing times of all boats are iid Gaussian random variables with expected value 35 minutes and standard deviation 5 minutes.
(a) What is the probability that the winning boat will finish the race in less than 25 minutes?
(b) What is the probability that the last boat will cross the finish line in more than 50 minutes? (c) Given this model, what is the probability that a boat will finish before it starts (negative finishing time)?