# Statistics Expected Value

The expected value (also known as the mathematical expectation or the mean) is a fundamental concept in probability theory and statistics. It represents the average or expected outcome of a random variable.

The expected value of a discrete random variable X is calculated as the weighted average of all possible outcomes of X, where the weights are the probabilities of those outcomes. Mathematically, it is represented as:

E(X) = ∑x_i * P(x_i)

where x_i represents the possible values of X, and P(x_i) is the corresponding probability of x_i.

For continuous random variables, the expected value is calculated as an integral, rather than a sum:

E(X) = ∫x * f(x) dx

where f(x) is the probability density function of X.

The expected value has important properties and applications in many fields, including economics, finance, engineering, and many others. It is used as a measure of central tendency, to make predictions about future outcomes, and as a basis for making decisions under uncertainty.