5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
2:20 minutesProblem 5.1.7
Textbook Question
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.
Verified step by step guidance1
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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