From Algorithms to Z-Scores - matloff - University of California, Davis

15.6. ESTIMATING THAT RELATIONSHIP FROM SAMPLE DATA 289
i ↓, j → 0 1 2 3
0 0.002 0.024 0.036 0.008
1 0.162 0.073 0.048 0.004
2 0.012 0.024 0.006 0.000
Now keep in mind that since mY ;B(t) is the conditional mean of Y given B, we need to use conditional
probabilities to compute it. For our example here of mY ;B(2), we need the probabilities
P (Y = k|B = 2). For instance,
P (Y = 1|B = 2) = pY,B(1, 2)
pB(2)
(15.2)
=
0.048
0.036 + 0.048 + 0.006
(15.3)
= 0.533 (15.4)
The other conditional P (Y = k|B = 2) are then found to be 0.400 for k = 0 and 0.067 for k = 2.
We then have
mY ;B(2) = 0.400 · 0 + 0.533 · 1 + 0.067 · 2 = 0.667 (15.5)
15.6 Estimating That Relationship from Sample Data
As noted, though, mW ;H(t) is a population function, dependent on population distributions. How
can we estimate this function from sample data?
Toward that end, let’s again suppose we have a random sample of 1000 people from Davis, with
(H1, W1), ..., (H1000, W1000) (15.6)
being their heights and weights. We again wish to use this data to estimate population values. But
the difference here is that we are estimating a whole function now, the whole curve mW ;H(t). That
means we are estimating infinitely many values, with one mW ;H(t) value for each t. 4 How do we
do this?
One approach would be as follows. Say we wish to find mW ;H(t) (note the hat, for “estimate of”!)
at t = 70.2. In other words, we wish to estimate the mean weight—in the population—among all
people of height 70.2. What we could do is look at all the people in our sample who are within, say,
1.0 inch of 70.2, and calculate the average of all their weights. This would then be our mW ;H(t).
4 Of course, the population of Davis is finite, but there is the conceptual population of all people who could live in
Davis.