Questions: Is the expected value of the probability distribution of a random variable always one of the possible values of x? Explain. Choose the correct answer below. A. No, because the expected value may not be a possible value of x for one trial, but it represents the average value of x over a large number of trials. B. Yes, because the expected value is the most likely outcome over a large number of trials. C. No, because the expected value is always 1. D. No, because the expected value is always 0.

Solution Steps

The expected value of a probability distribution is a measure of the center of the distribution. It represents the long-term average value of the random variable over many trials. Mathematically, for a discrete random variable \( X \) with possible values \( x_1, x_2, \dots, x_n \) and corresponding probabilities \( P(x_1), P(x_2), \dots, P(x_n) \), the expected value \( E(X) \) is calculated as: \[ E(X) = \sum_{i=1}^n x_i P(x_i). \]

The expected value is a weighted average of the possible values of \( X \), where the weights are the probabilities of each value. This means that the expected value does not necessarily have to be one of the possible values of \( X \). For example, consider a random variable \( X \) that takes the values 1 and 2 with equal probability: \[ E(X) = 1 \cdot 0.5 + 2 \cdot 0.5 = 1.5. \] Here, the expected value \( 1.5 \) is not one of the possible values of \( X \).

• A. This option correctly states that the expected value may not be a possible value of \( X \) for one trial but represents the average value over many trials. This aligns with our understanding.
• B. This option incorrectly claims that the expected value is the most likely outcome, which is not necessarily true.
• C. This option incorrectly states that the expected value is always 1, which is not the case.
• D. This option incorrectly states that the expected value is always 0, which is also not the case.

Final Answer