Identifying Probability Distributions In Exercises 27 and 28, 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: 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 individual probability must be between 0 and 1 inclusive.
2. Step 2: Verify the first condition by summing all the probabilities in the table. Add P(0), P(1), P(2), P(3), and P(4). Use the formula: \( \sum P(x) = P(0) + P(1) + P(2) + P(3) + P(4) \).
3. Step 3: Check the second condition by ensuring that each probability value in the table (0.30, 0.25, 0.25, 0.15, 0.05) is between 0 and 1 inclusive. If any value is outside this range, the distribution is not valid.
4. Step 4: If both conditions are satisfied, conclude that the given distribution is a probability distribution. If either condition fails, explain why the distribution is not valid.
5. Step 5: Interpret the results and provide reasoning based on the checks performed in steps 2 and 3. For example, if the sum of probabilities is not equal to 1, state the discrepancy and its impact on validity.

Key Concepts

Probability Distribution

A probability distribution describes how probabilities are assigned to each possible value of a random variable. For a distribution to be valid, the probabilities must be non-negative and sum to one. This concept is fundamental in statistics as it provides a framework for understanding the likelihood of different outcomes.

Sum of Probabilities

In a valid probability distribution, the sum of all probabilities must equal 1. This ensures that the total likelihood of all possible outcomes is accounted for. If the sum is less than or greater than 1, the distribution cannot be considered valid, indicating an error in the assignment of probabilities.

Non-negativity of Probabilities

Probabilities must be non-negative, meaning they cannot be less than zero. This principle is essential because negative probabilities do not have a meaningful interpretation in the context of likelihood. Each probability value in a distribution must reflect a real chance of occurrence, reinforcing the need for non-negativity.