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).

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For a valid probability distribution, the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1. In the given data, we need to check if these conditions are satisfied for the random variable x.

Mean of a Probability Distribution

The mean, or expected value, of a probability distribution is calculated by multiplying each value of the random variable by its corresponding probability and summing these products. This provides a measure of the central tendency of the distribution, indicating the average outcome if the experiment were repeated many times.

Standard Deviation of a Probability Distribution

The standard deviation measures the dispersion or spread of a probability distribution around its mean. It is calculated by taking the square root of the variance, which is the average of the squared differences between each value and the mean, weighted by their probabilities. A smaller standard deviation indicates that the values tend to be closer to the mean.