Textbook Question
Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.
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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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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Continuous Uniform Distribution
A continuous uniform distribution is a probability distribution where all outcomes in a specified range are equally likely. The distribution is defined by two parameters, a and b, which represent the minimum and maximum values. The probability density function (PDF) is constant between these limits, and the total area under the curve equals 1, indicating that the total probability of all outcomes is 100%.
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Probability Density Function (PDF)
The probability density function (PDF) describes the likelihood of a continuous random variable taking on a specific value. For a continuous uniform distribution, the PDF is a horizontal line, indicating that the probability is evenly distributed across the range. The height of the PDF is calculated as 1 divided by the range (b - a), ensuring that the area under the curve equals 1, which represents the total probability.
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Calculating Probability
To find the probability of a continuous random variable falling within a specific range, you calculate the area under the PDF over that interval. For a continuous uniform distribution, this is done by multiplying the height of the PDF by the width of the interval. For example, to find the probability that the waiting time is between 2 and 3 minutes, you would calculate the area of the rectangle formed by these limits, which is the height (0.2) times the width (1).
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