11 LINEAR MODELS AND LINEAR REGRESSION
The tools developed in Chapters 9 and 10 for parameter estimation and model
verification are applied in this chapter to a very useful class of models encoun-
tered in science and engineering. A commonly occurring situation is one in
which a random quantity, Y, is a function of one or more in dependent (and
deterministic) variables x 1 ,x 2 ,..., and xm. For example, wind load (Y) acting
on a structure is a function of height (x); the intensity (Y) of strong motion
earthquakes is dependent on the distance from the epicenter (x); housing price
(Y) is a function of lo cation (x 1 )andage(x 2 ); and chemical yield (Y) may be
related to temperature (x 1 ), pressure (x 2 ), and acid content (x 3 ).
Given a sample of Y values with their associated values of xi, i 1, 2,... , m,
we are interested in estimating on the basis of this sample the relationship
between Y and the independent variables x 1 ,x 2 ,..., and xm. In what follows,
we concentrate on some simple cases of the broadly defined problem stated
above.
11.1 Simple Linear R egression
We assume in this section that random variable Y is a function of only one
independent variable and that their relationship is linear. By a linear relation-
sh ip we mean that the mean of Y , E Y , is known to be a linear function of x,
that is,
The two constants, intercept and slope , are unknown and are to be
estimated from a sample of Y values with their associated values of x. Note
Fundamentals of Probability and Statistics for Engineer sT.T. Soong 2004 John Wiley & Sons, Ltd
ISBN s: 0-470-86813-9 (H B) 0-470-86814-7 (PB)
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