Let a be a number between 0 and 1, and define a sequence W1, W2, W3,... by W0 = µ Wt = αXÌ…t +(1-α)Wt-1 for t = 1,2 , . . . Substituting for Wt-1 its representation in terms of XÌ…t-1 and Wt-2, then substitutin for Wt-2, and so on, result in
Wt = aXÌ…t + a(1 - a)XÌ…t-1 + . . . + α(1 - α)t-1XÌ…1+ (1 - α)tµ
The fact that Wt depends not only on XÌ…t 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 XÌ…t's.
a. Show that E(Wt) = µ.
b. Let σt2 = V(Wt), and show that

c. An exponentially weighted moving-average control chart plots the Wt's and uses control limits µ0 ± 3σt (or xÌ¿ in place of µ0). Construct such a chart for the data of Example 16.9, using µ0 = 40.

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