"True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


The mean of the random variable of a probability distribution describes how the outcomes vary."

Verified step by step guidance

1. Step 1: Understand the concept of the mean in a probability distribution. The mean (also called the expected value) of a random variable in a probability distribution represents the central tendency or the average value of the outcomes, weighted by their probabilities.
2. Step 2: Clarify the statement provided. The statement claims that the mean describes how the outcomes vary. This is incorrect because the mean does not describe variability; it describes the central location of the distribution.
3. Step 3: Introduce the correct term for variability. The measure that describes how the outcomes vary is the variance or standard deviation, not the mean. Variance quantifies the spread of the outcomes around the mean.
4. Step 4: Rewrite the statement as a true statement. A correct version of the statement would be: 'The mean of the random variable of a probability distribution describes the central tendency of the outcomes, not their variability.'
5. Step 5: Summarize the distinction between mean and measures of variability. The mean provides information about the average outcome, while measures like variance and standard deviation provide information about the spread or variability of the outcomes.

Key Concepts

Mean of a Probability Distribution

The mean of a probability distribution, also known as the expected value, represents the average outcome of a random variable when considering all possible values weighted by their probabilities. It provides a central point around which the values of the random variable are distributed, but it does not indicate how much the outcomes vary.

Variance and Standard Deviation

Variance measures the spread of a set of values in a probability distribution, indicating how much the outcomes differ from the mean. The standard deviation, the square root of variance, provides a more interpretable measure of variability, showing the average distance of each outcome from the mean. Together, these concepts help describe the distribution's variability.

Random Variable

A random variable is a numerical outcome of a random phenomenon, which can be discrete (taking specific values) or continuous (taking any value within a range). Understanding random variables is crucial for analyzing probability distributions, as they form the basis for calculating probabilities, means, and variances.