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Undoubtedly, the most widely used model for the distribution of a random variable is a normal distribution. 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 engineer 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 conclusions 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 infinitesimal effects, such as temperature and humidity drifts, vibrations, cutting angle variations, cutting tool wear, bearing wear, rotational speed variations, mounting and xturing 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 distribution 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 later.
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CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
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Figure 4-10 Normal probability density functions for selected values of the parameters and 2.
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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 E1X 2 determines the center of the probability density function and the value of 2 V1X 2 determines the width. Figure 4-10 illustrates several normal probability density functions with selected values of and 2. Each has the characteristic symmetric bell-shaped curve, but the centers and dispersions differ. The following de nition provides the formula for normal probability density functions. De nition 1 12
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A random variable X with probability density function f 1x2
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is a normal random variable with parameters , where Also, E1X 2 and V1X2
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and the notation N1 , 2 2 is used to denote the distribution. The mean and variance of X are shown to equal and 2, respectively, at the end of this Section 5-6.
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EXAMPLE 4-10
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Assume that the current measurements in a strip of wire follow a normal distribution with a mean of 10 milliamperes and a variance of 4 (milliamperes)2. What is the probability that a measurement exceeds 13 milliamperes Let X denote the current in milliamperes. The requested probability can be represented as P1X 132. This probability is shown as the shaded area under the normal probability density function in Fig. 4-11. Unfortunately, there is no closed-form expression for the integral of a normal probability density function, and probabilities based on the normal distribution are typically found numerically or from a table (that we will later introduce). Some useful results concerning a normal distribution are summarized below and in Fig. 4-12. For any normal random variable, P1 P1 P1 2 3 X X X 2 2 2 3 2 0.6827 0.9545 0.9973
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Also, from the symmetry of f 1x2, P1X 2 P1X 2 0.5. Because f(x) is positive for all x, this model assigns some probability to each interval of the real line. However, the
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