Introduction to Probability and Statistics for Engineers and Scientists

13.6Other Control Charts for Detecting Changes in the Population Mean 567

where the foregoing used the fact thatW 0 =μ. Thus we see from Equation 13.6.2 that
Wtis a weighted average of all the subgroup averages up to timet, giving weightαto
the most recent subgroup and then successively decreasing the weight of earlier subgroup
averages by the constant factor 1−α, and then giving weight (1−α)tto the in-control
population mean.
The smaller the value ofα, the more even the successive weights. For instance, if
α=.1 then the initial weight is .1 and the successive weights decrease by the factor .9;
that is, the weights are .1, .09, .081, .073, .066, .059, and so on. On the other hand,
if one chooses, say,α=.4, then the successive weights are .4, .24, .144, .087, .052,...
Since the successive weightsα(1−α)i−^1 ,i=1, 2,..., can be written as

α(1−α)i−^1 =αe−βi

where

α=

α

1 −α

, β=−log(1−α)

we say that the successively older data values are “exponentially weighted” (see
Figure 13.4).
To compute the mean and variance of theWt, recall that, when in control, the subgroup
averagesXiare independent normal random variables each having meanμand variance
σ^2 /n. Therefore, using Equation 13.6.2, we see that

E[Wt]=μ[α+α(1−α)+α(1−α)^2 +···+α(1−α)t−^1 +(1−α)t]

=

μα[ 1 −(1−α)t]

1 −(1−α)

+μ(1−α)t

=μ

i

0.4

0.3

0.2

0.1

0.00246 810

a(1

−^ a

i − )

1

FIGURE 13.4 Plot ofα(1−α)i−^1 whenα=.4.