Figure 2.7 illustrates how, when the relationship between Y and X is nonlinear in an interval, the average value of Y in the interval can be a biased estimate of E (Y|x) at the center of the interval....

Figure 2.7 illustrates how, when the relationship between Y and X is nonlinear in an interval, the average value of Y in the interval can be a biased estimate of E (Y|x) at the center of the interval. Imagine that X-values are evenly distributed in an interval centered at x_i, and let µ_i
= E (Y|x_i).

(a) If the relationship between Y and X is linear in the interval, is the average value of Y a biased or an unbiased estimator of µ_i?

(b) Are there any circumstances under which the average Y in the interval is an unbiased estimator of µi if the relationship between Y and X is nonlinear in the interval?

(c) What happens when the distribution of X-values in the interval is not uniform?