In this chapter, we will analyze probability statements about random variables that can take any value on a continuum. Measures of time, distance, or temperature fit naturally into this category. In such cases, the probability that the random variable takes a single specific value—for instance, the probability that a car will travel precisely 27.236 miles on a gallon of gasoline—is 0. It is also convenient to regard as continuous essentially discrete random variables that are measured on such a fine grid that the probability of occurrence of any specific value is trivially small. For instance, in a study of the total debt incurred by students while attending four-year colleges, it is certainly true that the total debt for any given student will be some integer number of cents. However, the probability that a randomly chosen student will have debts totaling precisely $9,274.57 is sufficiently small that the random variable of interest can be treated as if it were continuous.
Although the assessment of probabilities for individual values of continuous random variables is meaningless, we may well be interested in the probability that such a variable lies in some given range. For example, the probability that a car travels between 27 and 28 miles on a gallon of gasoline or the probability that a randomly chosen student has incurred debts between $9,000 and $10,000 may be useful quantities to evaluate. Therefore, in characterizing probability distributions for continuous random variables, a natural place to begin (by analogy with our discussion of discrete random variables) is with the idea of cumulative probability. This will provide us with a measure of the probability that a random variable does not exceed any specific value.
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