In Problems 9–14, determine whether the distribution is a discrete probability distribution. If not, state why.

Verified step by step guidance

1. Step 1: Understand the definition of a discrete probability distribution. It must satisfy two conditions: (a) each probability \(P(x)\) is between 0 and 1 inclusive, and (b) the sum of all probabilities equals 1.
2. Step 2: Check the probabilities given in the table: \(P(1) = 0\), \(P(2) = 0\), \(P(3) = 0\), \(P(4) = 0\), and \(P(5) = 1\). All these values are between 0 and 1, so the first condition is satisfied.
3. Step 3: Calculate the sum of all probabilities: \$0 + 0 + 0 + 0 + 1 = 1$. This satisfies the second condition that the total probability must be 1.
4. Step 4: Confirm that the values of \(x\) are discrete (individual points 1, 2, 3, 4, 5), which fits the requirement for a discrete distribution.
5. Step 5: Conclude that since both conditions are met and the \(x\) values are discrete, the given distribution is a valid discrete probability distribution.

Key Concepts

Discrete Probability Distribution

A discrete probability distribution assigns probabilities to distinct, countable outcomes of a random variable. Each probability must be between 0 and 1, and the sum of all probabilities must equal 1. This ensures the distribution accurately represents all possible outcomes.

Probability Rules

Probabilities must satisfy two main rules: no probability can be negative or greater than 1, and the total probability across all outcomes must sum to exactly 1. These rules guarantee that the distribution is valid and reflects a complete set of outcomes.

Interpreting Probability Tables

A probability table lists possible values of a random variable alongside their probabilities. To determine if it represents a valid discrete distribution, check that all probabilities are valid and sum to 1. For example, a table with one outcome having probability 1 and others 0 is valid.