In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).



Does the table describe a probability distribution?

Verified step by step guidance

1. Step 1: Understand the requirements for a probability distribution. A probability distribution must satisfy two conditions: (1) The sum of all probabilities must equal 1, and (2) each probability value must be between 0 and 1 inclusive.
2. Step 2: Examine the table provided. The table lists values of x (the number of adults reporting sleepwalking) and their corresponding probabilities P(x). Verify that all P(x) values are between 0 and 1.
3. Step 3: Add up all the probabilities in the table: P(0) + P(1) + P(2) + P(3) + P(4) + P(5). Use the values provided in the table: 0.172 + 0.363 + 0.306 + 0.129 + 0.027 + 0.002.
4. Step 4: Check if the sum of the probabilities equals 1. If the sum is exactly 1, the first condition for a probability distribution is satisfied.
5. Step 5: Confirm that all probabilities are between 0 and 1 inclusive. If both conditions are satisfied, then the table describes a probability distribution.

Key Concepts

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. It provides a complete description of the likelihood of each outcome occurring. In this case, the table lists the number of adults who reported sleepwalking (x) and their corresponding probabilities (P(x)), which collectively form a discrete probability distribution.

Properties of Probability

For a set of probabilities to represent a valid probability distribution, two key properties must be satisfied: each probability must be between 0 and 1, and the sum of all probabilities must equal 1. In the provided table, we can check these properties to determine if the data indeed forms a valid probability distribution.

Discrete Random Variable

A discrete random variable is a variable that can take on a countable number of distinct values, often representing counts or categories. In this context, the variable x represents the number of adults in groups of five who reported sleepwalking, making it a discrete random variable. Understanding this concept is essential for analyzing the associated probability distribution.