Let α be a number between 0 and 1, and define a sequence for t = 1, 2,……. Substituting...

Let α be a number between 0 and 1, and define a sequence for t = 1, 2,……. Substituting for its representation in terms of , 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 ( in place of μ0). Construct such a chart for the data of Example 16.9, using μ₀ =40