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