Page 141 - Applied statistics and probability for engineers

Section 4-6/Normal Distribution 119

4-6 Normal Distribution
Undoubtedly, the most widely used model for a continuous measurement is a normal ran-
dom variable. Whenever a random experiment is replicated, the random variable that equals
the average (or total) result over the replicates tends to have a normal distribution as the
number of replicates becomes large. De Moivre presented this fundamental result, known
as the central limit theorem, in 1733. Unfortunately, his work was lost for some time, and
Gauss independently developed a normal distribution nearly 100 years later. Although De
Moivre was later credited with the derivation, a normal distribution is also referred to as a
Gaussian distribution.
When do we average (or total) results? Almost always. For example, an automotive engi-
neer may plan a study to average pull-off force measurements from several connectors. If we
assume that each measurement results from a replicate of a random experiment, the normal
distribution can be used to make approximate conclusions about this average. These conclu-
sions are the primary topics in the subsequent chapters of this book.
Furthermore, sometimes the central limit theorem is less obvious. For example, assume that
the deviation (or error) in the length of a machined part is the sum of a large number of inini-
tesimal effects, such as temperature and humidity drifts, vibrations, cutting angle variations,
cutting tool wear, bearing wear, rotational speed variations, mounting and ixture variations,
variations in numerous raw material characteristics, and variation in levels of contamination.
If the component errors are independent and equally likely to be positive or negative, the total
error can be shown to have an approximate normal distribution. Furthermore, the normal dis-
tribution arises in the study of numerous basic physical phenomena. For example, the physicist
Maxwell developed a normal distribution from simple assumptions regarding the velocities
of molecules.
The theoretical basis of a normal distribution is mentioned to justify the somewhat complex
form of the probability density function. Our objective now is to calculate probabilities for a
normal random variable. The central limit theorem will be stated more carefully in Chapter 5.
Random variables with different means and variances can be modeled by normal probability
density functions with appropriate choices of the center and width of the curve. The value of
2
E X ( ) = μ determines the center of the probability density function, and the value of V X ( ) = σ
determines the width. Figure 4-10 illustrates several normal probability density functions with
2
selected values of μ and σ . Each has the characteristic symmetric bell-shaped curve, but the cen-
ters and dispersions differ. The following deinition provides the formula for normal probability
density functions.

Normal Distribution
A random variable X with probability density function
− ( x − ) μ 2
f x ( ) = 1 e 2σ 2 − ∞ < x < ∞ (4-8)
2πσ
μ
is a normal random variable with parameters μ where −∞ < < ∞, and σ > 0. Also,
E X ( ) = μ and V X ( ) = σ 2 (4-9)
and the notation N μ ( ,È ) is used to denote the distribution.
2

2
The mean and variance of X are shown to equal μ and σ , respectively, in an exercise at the
end of Chapter 5.