Copyright © Cengage Learning. All rights reserved. 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

Copyright © Cengage Learning. All rights reserved. 8.5 The Normal Distribution

3 Probability Density Functions

4 The probability distributions with finite random variables—that is, random variables that take on finitely many values are referred to as finite probability distributions. In this section, we consider probability distributions associated with a continuous random variable—that is, a random variable that may take on any value lying in an interval of real numbers. Such probability distributions are called continuous probability distributions.

5 Probability Density Functions Unlike a finite probability distribution, which may be exhibited in the form of a table, a continuous probability distribution is defined by a function f whose domain coincides with the interval of values taken on by the random variable associated with the experiment. Such a function f is called the probability density function associated with the probability distribution, and it has the following properties: 1. f (x) is nonnegative for all values of x in its domain.

6 Probability Density Functions 2. The area of the region between the graph of f and the x-axis is equal to 1 (Figure 13). Figure 13 A probability density function

7 Probability Density Functions Now suppose we are given a continuous probability distribution defined by a probability density function f. Then the probability that the random variable X assumes a value in an interval a < x < b is given by the area of the region between the graph of f and the x-axis from x = a to x = b (Figure 14). Figure 14 P(a < X < b) is given by the area of the shaded region

8 Probability Density Functions We denote the value of this probability by P(a < X < b). Observe that property 2 of the probability density function states that the probability that a continuous random variable takes on a value lying in its range is 1, a certainty, which is expected.

9 Normal Distributions

10 Normal Distributions The mean  and the standard deviation  of a continuous probability distribution have roughly the same meanings as the mean and standard deviation of a finite probability distribution. Thus, the mean of a continuous probability distribution is a measure of the central tendency of the probability distribution, and the standard deviation of the probability distribution measures its spread about its mean.

11 Normal Distributions Both of these numbers will play an important role in the following discussion. For the remainder of this section, we will discuss a special class of continuous probability distributions known as normal distributions. Normal distributions are without a doubt the most important of all the probability distributions.

12 Normal Distributions Many phenomena—such as the heights of people in a given population, the weights of newborn infants, the IQs of college students, the actual weights of 16-ounce packages of cereals, and so on—have probability distributions that are normal. The normal distribution also provides us with an accurate approximation to the distributions of many random variables associated with random-sampling problems.

13 Normal Distributions The graph of a normal distribution, which is bell shaped, is called a normal curve (Figure 15). A normal curve Figure 15

14 Normal Distributions The normal curve (and therefore the corresponding normal distribution) is completely determined by its mean  and standard deviation . In fact, the normal curve has the following characteristics, which are described in terms of these two parameters. 1. The curve has a peak at x = . 2. The curve is symmetric with respect to the vertical line x = .

15 Normal Distributions 3. The curve always lies above the x-axis but approaches the x-axis as x extends indefinitely in either direction. 4. The area under the curve is For any normal curve, 68.27% of the area under the curve lies within 1 standard deviation of the mean (that is, between  –  and  +  ), 95.45% of the area lies within 2 standard deviations of the mean, and 99.73% of the area lies within 3 standard deviations of the mean.

16 Normal Distributions Figure 16 shows two normal curves with different means  1 and  2 but the same standard deviation. Figure 16 Two normal curves that have the same standard deviation but different means

17 Normal Distributions Next, Figure 17 shows two normal curves with the same mean but different standard deviations  1 and  2. (Which number is smaller?) Figure 17 Two normal curves that have the same mean but different standard deviations

18 Normal Distributions The mean  of a normal distribution determines where the center of the curve is located, whereas the standard deviation  of a normal distribution determines the peakedness (or flatness) of the curve. As this discussion reveals, there are infinitely many normal curves corresponding to different choices of the parameters  and  that characterize such curves.

19 Normal Distributions Fortunately, any normal curve may be transformed into any other normal curve, so in the study of normal curves it suffices to single out one such particular curve for special attention. The normal curve with mean  = 0 and standard deviation  = 1 is called the standard normal curve. The corresponding distribution is called the standard normal distribution. The random variable itself is called the standard normal random variable and is commonly denoted by Z.