Expected value

Expected value is a fundamental concept in probability theory. It serves as a generalization of the weighted average. You can think of it as a way to find the center of a set of possible outcomes. This value tells us what we might expect to happen on average over many trials. It is known by many names, including expectation, expectancy, and the first moment.

In mathematics, it provides a single number that represents the long-term average of a random variable.

The mechanism of expected value depends on the type of outcomes involved. For a random variable with a finite number of outcomes, the expected value is a weighted average. You take each possible outcome and multiply it by its probability of occurring. Then, you add all those results together. This process accounts for the fact that some outcomes are more likely than others. If every outcome is equally likely, the expected value becomes a standard arithmetic mean.

Mathematicians categorize expected value into different stages of complexity. The simplest stage involves finitely many outcomes, like flipping a coin. The next stage involves countably infinite outcomes, where there are endless possibilities that can still be listed. For these infinite sets, the expected value is an infinite sum of outcomes multiplied by their probabilities. A third stage involves continuous random variables. These variables do not have a list of outcomes but instead follow a probability density function. In these cases, the expected value is found using integration to measure the area under a curve.

The history of this idea began in the mid-17th century. It emerged from the "problem of points," a puzzle about dividing stakes fairly when a game ends early. In 1654, the Chevalier de Méré presented this problem to Blaise Pascal. Pascal began a famous correspondence with Pierre de Fermat to solve it. Both men independently arrived at the same solution. They used the principle that the value of a future gain should be proportional to the chance of receiving it. They did not publish their work, but they shared it with friends in Paris.

Following Pascal and Fermat, Christiaan Huygens expanded the theory significantly. After visiting Paris, the Dutch mathematician published "De ratiociniis in ludo aleæ" in 1657. His treatise was the first successful attempt to lay the foundations of probability theory. He added rules for calculating expectations in complex games with three or more players. Later, in 1814, Pierre-Simon Laplace published a work that explicitly defined the concept. By the mid-19th century, Pafnuty Chebyshev began to study random variables and their expectations systematically. The notation using the letter E became popular after W. A. Whitworth used it in 1901.

We can see the power of expected value in specific, real-world examples. Consider a fair six-sided die where each number has a probability of 1/6. The expected value of a single roll is 3.5. While you can never actually roll a 3.5, the average of many rolls will converge to this number.

Another example is American roulette. A $1 bet on a single number pays $35 if you win, but you lose your dollar if you do not. The probability of winning is 1/38. The expected profit for this bet is -$0.05. This means that over 190 bets, a player would likely lose about $10.

In advanced mathematics, the concept is unified through measure theory. The expected value is formally defined using the Lebesgue integral. This provides an axiomatic foundation that works for all types of random variables. This high-level definition allows mathematicians to handle multidimensional variables, such as random vectors or matrices. It also clarifies how to work with absolutely continuous random variables. By using these tools, the theory of expected value connects simple counting to complex systems in physics and advanced statistics.