Compute the correlation coefficient r for X and Y of Example 5.16 (the covariance has al...

Compute the correlation coefficient r for X and Y of Example 5.16 (the covariance has already been computed).

Reference example 5.16

The joint and marginal pdf’s of X = amount of almonds and Y = amount of cashews were

It might appear that the relationship in the insurance example is quite strong since Cov(X, Y ) = 1875, whereas in the nut example would seem to imply quite a weak relationship. Unfortunately, the covariance has a serious defect that makes it impossible to interpret a computed value. In the insurance example, suppose we had expressed the deductible amount in cents rather than in dollars. Then 100X would replace X, 100Y would replace Y, and the resulting covariance would be Cov(100X, 100Y) = (100)(100)Cov(X, Y) = 18,750,000. If, on the other hand, the deductible amount had been expressed in hundreds of dollars, the computed covariance would have been (.01)(.01)(1875) = .1875. The defect of covariance is that its computed value depends critically on the units of measurement. Ideally, the choice of units should have no effect on a measure of strength of relationship. This is achieved by scaling the covariance.