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Probability distribution is a fundamental concept in statistics, especially when dealing with random variables. It provides a list of all the possible values a random variable can take, along with the probability associated with each value. In our exercise, the random variable is the number of cars owned by an American household. The probability distribution is demonstrated in a table, presenting various numbers of cars from 0 to 5, along with the likelihood of each occurrence. Probability distributions are crucial because they help us understand how a random variable behaves. When examining them, ensure that all probabilities add up to 1, as they represent the total certainty of all possible outcomes. This trait is evident in the distribution outlined, where each probability, when summed, equals 1. Additionally, probability distributions can be of different types, such as discrete or continuous, depending on whether the random variables are countable or not.

A discrete random variable is a type of variable that can take on a countable number of specific values. For instance, in our current scenario, the random variable is the number of cars a household owns, which can specifically be 0, 1, 2, 3, 4, or 5. This contrasts with continuous random variables, which can assume an infinite number of values within a range. Understanding discrete random variables is important as they often relate to real-world integers. To fully comprehend them, visualize scenarios where only a limited set of outcomes is possible. Discrete random variables help in establishing structured and countable outcomes, making calculations involving probability distributions more manageable.

The mean of distribution, often referred to as the expected value, offers insight into the central tendency of a random variable's probability distribution. It is essentially the average or the expected outcome you can anticipate if you were to repeat a random experiment multiple times.To compute the mean of a probability distribution for a discrete random variable, multiply each possible value the random variable can take by its corresponding probability, and then sum all these products. This approach reflects the formula: \[ \text{Mean (Expected Value)} = E(X) = \sum (x_i \cdot p_i) \]where \(x_i\) is a value that the random variable can assume, and \(p_i\) is the probability of \(x_i\).Using the exercise example: - The calculation involves multiplying each number of cars by its probability and then adding the results: \[ E(X) = 0 \cdot 0.09 + 1 \cdot 0.36 + 2 \cdot 0.35 + 3 \cdot 0.13 + 4 \cdot 0.05 + 5 \cdot 0.02 = 1.75 \] The result, 1.75, represents the expected number of cars owned in an American household, showcasing how the mean of a distribution summarizes the variable’s trend over time.