Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

Verified step by step guidance

1. Step 1: Recall the conditions for a probability distribution. A probability distribution must satisfy two criteria: (1) Each probability value, p(x), must be between 0 and 1 inclusive, and (2) The sum of all probability values must equal 1.
2. Step 2: Check the first condition. Inspect each value of p(x) in the table to ensure that all values are between 0 and 1 inclusive. Specifically, verify that 0 ≤ p(x) ≤ 1 for each value.
3. Step 3: Check the second condition. Add all the values of p(x) from the table: 0.03 + 0.09 + 0.19 + 0.32 + 0.37. Ensure that the sum equals 1.
4. Step 4: If any p(x) value is outside the range [0, 1], or if the sum of p(x) values does not equal 1, then the distribution is not a probability distribution. Identify and explain the specific issue.
5. Step 5: If both conditions are satisfied, conclude that the given distribution is a valid probability distribution.

Key Concepts

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For a discrete random variable, it assigns a probability to each possible value, ensuring that the sum of all probabilities equals 1. This concept is fundamental in determining whether a given set of probabilities constitutes a valid probability distribution.

Properties of Probability

The properties of probability state that each probability must be between 0 and 1, inclusive, and the total probability of all outcomes must sum to 1. If any probability is negative or if the total does not equal 1, the distribution is invalid. Understanding these properties is essential for evaluating whether the provided probabilities form a legitimate probability distribution.

Discrete Random Variables

Discrete random variables are those that can take on a countable number of distinct values, often represented in a table format. Each value has an associated probability, and the analysis of these variables often involves checking the validity of their probability distributions. Recognizing the nature of discrete random variables is crucial for interpreting the data presented in the question.