Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.
Plane Crashes The table lists causes of fatal plane crashes with their corresponding probabilities.

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Step 1: Verify if the given data represents a probability distribution. To do this, check two key requirements: (a) All probabilities must be between 0 and 1, and (b) The sum of all probabilities must equal 1.

Step 2: Add the probabilities provided in the table: Pilot Error (0.58), Mechanical (0.17), Weather (0.06), Sabotage (0.09), and Other (0.10). Use the formula: \( \text{Sum} = P_1 + P_2 + P_3 + P_4 + P_5 \).

Step 3: If the sum of probabilities equals 1 and all probabilities are between 0 and 1, confirm that the data represents a probability distribution. If not, identify which requirement is violated.

Step 4: To find the mean of the probability distribution, use the formula \( \mu = \sum (x \cdot P(x)) \), where \( x \) represents the causes (numerical values assigned to each category) and \( P(x) \) represents the probabilities.

Step 5: To find the standard deviation, use the formula \( \sigma = \sqrt{\sum (x^2 \cdot P(x)) - \mu^2} \). Calculate \( \sum (x^2 \cdot P(x)) \), subtract \( \mu^2 \), and take the square root of the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. For a set of outcomes to qualify as a probability distribution, the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1. In the context of the plane crash causes, we need to verify if these conditions are met.

Mean of a Probability Distribution

The mean of a probability distribution, also known as the expected value, is a measure of the central tendency of the distribution. It is calculated by multiplying each outcome by its probability and summing these products. In this case, if the distribution is valid, we would compute the mean to understand the average cause of plane crashes based on the given probabilities.

Standard Deviation of a Probability Distribution

The standard deviation of a probability distribution quantifies the amount of variation or dispersion of the outcomes. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. Understanding the standard deviation helps assess the reliability of the probabilities associated with the causes of plane crashes.

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