Let a be a number between 0 and 1, and define a sequence by and for . Substituting for its representation in terms of and , then substituting for , and so on, results in The fact that Wt depends not only on but also on averages for past time points, albeit with (exponentially) decreasing weights, suggests that changes in the process mean will be more quickly reflected in the Wt

’s than in the individual ’s.

a. Show that .

b. Let , and show that

c. An exponentially weighted moving-average control chart plots the Wt

’s and uses control limits (or in place of m0). Construct such a chart for the data of Example 16.9, using .