5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
2:32 minutesProblem 5.1.9
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.
Online Courses College students are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they take one or more online courses (based on data from Sallie Mae).
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Step 1: Verify if the given data represents a probability distribution. To do this, check two conditions: (a) All probabilities P(x) must be between 0 and 1, inclusive, and (b) The sum of all probabilities P(x) must equal 1.
Step 2: Calculate the sum of the probabilities P(x) provided in the table: \( P(0) + P(1) + P(2) + P(3) \). Use the values \( 0.104, 0.351, 0.396, \text{and } 0.149 \) from the table.
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 not satisfied.
Step 4: To find the mean (expected value) of the probability distribution, use the formula \( \mu = \sum [x \cdot P(x)] \). Multiply each value of \( x \) by its corresponding \( P(x) \), then sum the results.
Step 5: To find the standard deviation, use the formula \( \sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]} \). First, calculate \( (x - \mu)^2 \) for each \( x \), multiply by \( P(x) \), sum the results, and then take the square root.
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