What does the expected value of a random variable represent?

The expected value of a random variable is a fundamental concept in probability and statistics, representing the long-term average or mean of all possible values of that variable. Essentially, it is calculated by multiplying each possible outcome by its probability and then summing all these products. This gives a single value that reflects the average outcome one would expect if the random variable were observed many times over.

This concept is crucial because it provides a summary measure of the entire distribution of values that a random variable can take, helping to understand what value to anticipate in the long run. It is not simply a range of values, the maximum value, or a count of frequencies but rather the central tendency around which the outcomes tend to cluster when observing the variable repeatedly. Thus, acknowledging the expected value allows for informed decision-making based on the probabilities and outcomes associated with random variables in various applications.