How do you calculate the expected value of a discrete random variable?

How do you calculate the expected value of a discrete random variable?

• A. By taking the average of all possible outcomes
• B. By summing the probabilities of each outcome
• C. By multiplying each possible value by its probability and summing
• D. By determining the most likely outcome only

Answer: (C)

The expected value of a discrete random variable is calculated by multiplying each possible value of the variable by its corresponding probability and then summing these products. This method provides a weighted average that takes into account not only the values that the random variable can assume but also how likely each of those values is to occur. For instance, if a discrete random variable can take on several values, the expected value formula is: \[ E(X) = \sum (x_i \cdot P(x_i)) \] where \( x_i \) represents each possible value and \( P(x_i) \) is the probability associated with \( x_i \). This computation gives you a single value that represents the center of the distribution of the random variable, providing a significant summary measure of its behavior. This approach contrasts with simply taking the average of all possible outcomes without consideration of their probabilities, which can lead to misleading results, especially when some outcomes are far more likely than others. Additionally, summing the probabilities alone does not yield an expected value, as probabilities must be weighted by the values they correspond to. Lastly, only determining the most likely outcome ignores the breadth of the distribution and does not give a comprehensive measure of the random variable's behavior.

The expected value of a discrete random variable is calculated by multiplying each possible value of the variable by its corresponding probability and then summing these products. This method provides a weighted average that takes into account not only the values that the random variable can assume but also how likely each of those values is to occur.

For instance, if a discrete random variable can take on several values, the expected value formula is:

[ E(X) = \sum (x_i \cdot P(x_i)) ]

where ( x_i ) represents each possible value and ( P(x_i) ) is the probability associated with ( x_i ). This computation gives you a single value that represents the center of the distribution of the random variable, providing a significant summary measure of its behavior.

This approach contrasts with simply taking the average of all possible outcomes without consideration of their probabilities, which can lead to misleading results, especially when some outcomes are far more likely than others. Additionally, summing the probabilities alone does not yield an expected value, as probabilities must be weighted by the values they correspond to. Lastly, only determining the most likely outcome ignores the breadth of the distribution and does not give a comprehensive measure of the random variable's behavior.