Continuous Random Variables
First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using -score transformations. Finally, it defines tail probabilities and illustrates how to find them.
Probability Computations for General Normal Random Variables
- To learn how to compute probabilities related to any normal random variable.
If is any normally distributed normal random variable then Figure 12.2 "Cumulative Normal Probability" can also be used to compute a probability of the form by means of the following equality.
If is a normally distributed random variable with mean and standard deviation , then
where denotes a standard normal random variable. can be any decimal number or can be any decimal number or .
The new endpoints and are the -scores of and as defined in Section 2.4.2 in Chapter 2 "Descriptive Statistics".
Figure 5.14 "Probability for an Interval of Finite Length" illustrates the meaning of the equality geometrically: the two shaded regions, one under the density curve for and the other under the density curve for , have the same area. Instead of drawing both bell curves, though, we will always draw a single generic bell-shaped curve with both an -axis and a -axis below it.
Figure 5.14 Probability for an Interval of Finite Length
Let be a normal random variable with mean and standard deviation . Compute the following probabilities.a. .
a. See Figure 5.15 "Probability Computation for a General Normal Random Variable".
a. See Figure 5.16 "Probability Computation for a General Normal Random Variable".
Figure 5.16 Probability Computation for a General Normal Random Variable
The lifetimes of the tread of a certain automobile tire are normally distributed with mean 37,500 miles and standard deviation 4,500 miles. Find the probability that the tread life of a randomly selected tire will be between 30,000 and 40,000 miles.
Let denote the tread life of a randomly selected tire. To make the numbers easier to work with we will choose thousands of miles as the units. Thus , and the problem is to compute . Figure 5.17 "Probability Computation for Tire Tread Wear" illustrates the following computation:
Figure 5.17 Probability Computation for Tire Tread Wear
Note that the two -scores were rounded to two decimal places in order to use Figure 12.2 "Cumulative Normal Probability".
Scores on a standardized college entrance examination (CEE) are normally distributed with mean 510 and standard deviation 60. A selective university considers for admission only applicants with CEE scores over 650. Find percentage of all individuals who took the CEE who meet the university's CEE requirement for consideration for admission.
Let denote the score made on the CEE by a randomly selected individual. Then is normally distributed with mean 510 and standard deviation 60. The probability that lie in a particular interval is the same as the proportion of all exam scores that lie in that interval. Thus the solution to the problem is , expressed as a percentage. Figure 5.18 "Probability Computation for Exam Scores" illustrates the following computation:
Figure 5.18 Probability Computation for Exam Scores
The proportion of all CEE scores that exceed 650 is , hence or about do.