Let  be a number between 0 and 1 and define a sequence W1, W2, W3, . . . by W0
and Wt
Xt (1  )Wt1 for t

1, 2, . . . . Substituting for Wt1 its representation in terms of Xt1 and Wt2, then substituting for Wt2, and so on, results in Wt
Xt (1  )Xt1

(1  )t1 X1 (1  )t The fact that Wt depends not only on Xt 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 Wts than in the individual Xts.

a. Show that E(Wt

)
.

b. Let 2 t
V(Wt

), and show that 2

t


[1 

2

(



1 



)2t

]


n 2



c. An exponentially weighted moving-average control chart plots the Wts and uses control limits 0  3t (or x in place of 0). Construct such a chart for the data of Example 16.9, using 0
40.