Introduction to Probability and Statistics for Engineers and Scientists

9.8Weighted Least Squares 389

50

45

40

35

30

25

20

15

Travel time

02 46 810

Distance (miles)

FIGURE 9.12 Example 9.8b.

The rationale behind this claim is that it seems reasonable to suppose thatY has
approximately a Poisson distribution. This is so since we can imagine that each of thex
cars will have a small probability of causing an accident and so, for largex, the number
of accidents should be approximately a Poisson random variable. Since the variance of
a Poisson random variable is equal to its mean, we see that

Var(Y)E[Y] sinceYis approximately Poisson

=α+βx

βx for largex ■

REMARKS

(a)Another technique that is often employed when the variance of the response depends
on the input level is to attempt to stabilize the variance by an appropriate transformation.
For example, ifYis a Poisson random variable with meanλ, then it can be shown [see
Remark (b)] that

√
Yhas approximate variance .25 no matter what the value ofλ. Based
on this fact, one might try to modelE[

√
Y]as a linear function of the input. That is, one
might consider the model

√

Y=α+βx+e