Page 352 - Fundamentals of Probability and Statistics for Engineers

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 independent (and
deterministic) variables x 1 , x 2 ,..., and x m . 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 location (x 1 ) and age (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 x i , i ˆ 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 x m . In what follows,
we concentrate on some simple cases of the broadly defined problem stated
above.

11.1 SIMPLE LINEAR REGRESSION

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-
f g
ship we mean that the mean of Y , E Y , is known to be a linear function of x,
that is,

EfYgˆ ‡ x: 11:1†

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 EngineersT.T . Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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