Expected value, a cornerstone of statistics and probability, indicates the average outcome of repeated events. Despite its ubiquity in fields such as economics and decision-making, it doesn’t predict individual outcomes and can be skewed by outliers. Its broad applications necessitate considering ethical implications due to potential unequal impacts.

## Definition

Expected value is a fundamental concept in statistics and probability theory. It represents the average outcome of a series of events if those events were to be repeated a very large number of times.

## Calculation

Expected value is computed by multiplying each of the possible outcomes by the probability of that outcome occurring, then summing all these values. In other words, it’s the sum of all possible values each multiplied by the probability of its occurrence.

## In Probability

In the context of probability, the expected value is often referred to as the “population mean,” and it’s used to give a sense of the “average” outcome one might expect from a probability distribution.

## In Statistics

In statistical theory, expected values are often used to summarize the central tendency of a random variable. In these contexts, the expected value is a measure of the center of a probability distribution.

## Linear Properties

One of the key properties of expected value is that it is linear. This means that the expected value of the sum of random variables is equal to the sum of the expected values of those individual random variables.

## Application in Decision Making

Expected value is a crucial concept in economics, finance, and decision-making. It is often used to determine the most likely outcome in a series of events, such as the potential return on an investment.

## Influence of Outliers

While the concept of expected value provides a theoretical average, it’s important to note that in practice, individual outcomes may vary significantly. This can be particularly noticeable when the distribution of events has a few extreme outliers. In these cases, the median or mode may be a more appropriate measure of central tendency.

## Limitations

Expected value is based on the assumption that every outcome is independent and identically distributed. If these conditions are not met, the expected value may not provide an accurate summary of the data.