6. Normal Distribution and Continuous Random Variables
Probabilities & Z-Scores w/ Graphing Calculator
2:20 minutesProblem 6.1.7
Textbook Question
Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Between 2 minutes and 3 minutes
Verified step by step guidance1
Step 1: Recognize that the problem involves a continuous uniform distribution. The graph provided shows a uniform distribution where the probability density function (PDF) is constant at P(x) = 0.2 over the interval [0, 5]. The total area under the curve is 1, as required for a probability distribution.
Step 2: Recall the formula for the probability in a continuous uniform distribution. The probability of a random variable falling within a specific range [a, b] is given by: P(a ≤ X ≤ b) = (b - a) * f(x), where f(x) is the constant height of the PDF.
Step 3: Identify the range of interest. The problem asks for the probability that the waiting time is between 2 minutes and 3 minutes. Here, a = 2 and b = 3.
Step 4: Substitute the values into the formula. Use f(x) = 0.2 (from the graph) and calculate the width of the interval (b - a), which is (3 - 2). The formula becomes: P(2 ≤ X ≤ 3) = (3 - 2) * 0.2.
Step 5: Simplify the expression to find the probability. Multiply the width of the interval by the height of the PDF to determine the probability. This will give you the final result.
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