6-1 The Standard Normal Distribution 247 Fortunately, we won’t actually use Formula 6-1, but examining the right side of the equation reveals that any particular normal distribution is determined by two parameters: the population mean, m, and population standard deviation, s. (In Formula 6-1, x is a variable, p = 3.14159c and e = 2.71828c.) Once specific values are selected for m and s, Formula 6-1 is an equation relating x and y, and we can graph that equation to get a result that will look like Figure 6-1. And that’s about all we need to know about Formula 6-1! Uniform Distributions The major focus of this chapter is the concept of a normal probability distribution, but we begin with a uniform distribution so that we can see the following two very important properties: 1. The area under the graph of a continuous probability distribution is equal to 1. 2. There is a correspondence between area and probability, so probabilities can be found by identifying the corresponding areas in the graph using this formula for the area of a rectangle: Area = height * width FORMULA 6-1 y = e- 1 21 x-m s 22 s22p DEFINITION A continuous random variable has a uniform distribution if its values are equally spread over the range of possible values. The graph of a uniform distribution results in a rectangular shape. Density Curve The graph of any continuous probability distribution is called a density curve, and any density curve must satisfy the requirement that the total area under the curve is exactly 1. This requirement that the area must equal 1 simplifies probability problems, so the following statement is really important: Because the total area under any density curve is equal to 1, there is a correspondence between area and probability. Waiting Times for Airport Security EXAMPLE 1 During certain time periods at JFK airport in New York City, passengers arriving at the security checkpoint have waiting times that are uniformly distributed between 0 minutes and 5 minutes, as illustrated in Figure 6-2 on the next page. Refer to Figure 6-2 to see these properties: ■ All of the different possible waiting times are equally likely. ■ Waiting times can be any value between 0 min and 5 min, so it is possible to have a waiting time of 1.234567 min. ■ By assigning the probability of 0.2 to the height of the vertical line in Figure 6-2, the enclosed area is exactly 1. (In general, we should make the height of the vertical line in a uniform distribution equal to 1>range.) continued
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