Suppose that the two measurements from flea beetles in Example 5.10.2 have the bivariate normal distribution with μ1 = 201, μ2 = 118, σ1 = 15.2, σ2 = 6.6, and ρ = 0.64. Suppose that the same two measurements from a second species also have the bivariate normal distribution with μ1 = 187, μ2 = 131, σ1 = 15.2, σ2 = 6.6, and ρ = 0.64. Let (X1, X2) be a pair of measurements on a flea beetle from one of these two species. Let a1, a2 be constants.
a. For each of the two species, find the mean and standard deviation of a1X1 + a2X2. (The variances for the two species will be the same. How do you know that?)
b. Find a1 and a2 to maximize the ratio of the difference between the two means found in part (a) to the standard deviation found in part (a). There is a sense in which this linear combination a1X1 + a2X2 does the best job of distinguishing the two species among all possible linear combinations.