The expected value is a measure of the central tendency of a random variable. It can be thought of as the 'center of mass' of the probability distribution; it's where you would "expect" the value to be on average. It's determined through a weight sum of all possible values a random variable can take, with the weights being their probabilities.
Key points about expected values are:
- Expected value for a random variable \(X\) is generally noted as \(E(X)\).
- For a discrete random variable, \( E(X) = \sum x_i P(x_i) \).
- For continuous random variables, like in the Cauchy distribution, the expected value is an integral: \( E(X) = \int_{-\infty}^{\infty} x f(x) \, dx \).
- In the case of Cauchy distribution, the expected value, and indeed higher moments like variance, are undefined due to the way its tails behave.
The expected value provides a foundation for many statistical concepts, though it can be counterintuitive in certain distributions, such as the Cauchy distribution.