Expected Value
The expected value is a weighted average of all possible values in a data set.
Learning Objective

Compute the expected value and explain its applications and relationship to the law of large numbers
Key Points
 The expected value refers, intuitively, to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained.
 The intuitive explanation of the expected value above is a consequence of the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as the sample size grows to infinity.
 From a rigorous theoretical standpoint, the expected value of a continuous variable is the integral of the random variable with respect to its probability measure.
Terms

weighted average
an arithmetic mean of values biased according to agreed weightings

integral
the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed

random variable
a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die
Full Text
In probability theory, the expected value refers, intuitively, to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained. More formally, the expected value is a weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value.
The weights used in computing this average are the probabilities in the case of a discrete random variable (that is, a random variable that can only take on a finite number of values, such as a roll of a pair of dice), or the values of a probability density function in the case of a continuous random variable (that is, a random variable that can assume a theoretically infinite number of values, such as the height of a person).
From a rigorous theoretical standpoint, the expected value of a continuous variable is the integral of the random variable with respect to its probability measure. Since probability can never be negative (although it can be zero), one can intuitively understand this as the area under the curve of the graph of the values of a random variable multiplied by the probability of that value. Thus, for a continuous random variable the expected value is the limit of the weighted sum, i.e. the integral.
Simple Example
Suppose we have a random variable
The expected value of
This calculation can be easily generalized to more complicated situations. Suppose that a rich uncle plans to give you $2,000 for each child in your family, with a bonus of $500 for each girl. The formula for the bonus is:
What is your expected bonus?
We could have calculated the same value by taking the expected number of children and plugging it into the equation:
Expected Value and the Law of Large Numbers
The intuitive explanation of the expected value above is a consequence of the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as the sample size grows to infinity. More informally, it can be interpreted as the longrun average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary senseâ€”the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
Uses and Applications
To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller.
This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods.
The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a "good" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is typically considered good in this context if it is an unbiased estimatorâ€”that is, if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.
In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function such as a von NeumannMorgenstern utility function.
Key Term Reference
 Objective
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 arithmetic mean
 Appears in these related concepts: Which Average: Mean, Mode, or Median?, Applications of Statistics, and Exercises
 average
 Appears in these related concepts: Mean: The Average, Average Value of a Function, and Averages
 continuous random variable
 Appears in these related concepts: Expected Value and Standard Error, Continuous Probability Distributions, and The Correction Factor
 continuous variable
 Appears in these related concepts: Types of Variables, Variables, and Distributions
 datum
 Appears in these related concepts: Change of Scale, Controlling for a Variable, and Type I and II Errors
 density
 Appears in these related concepts: Density Calculations, Volume and Density, and The Density Scale
 dependent variable
 Appears in these related concepts: The Cartesian System, Converting between Exponential and Logarithmic Equations, and What is a Quadratic Function?
 discrete random variable
 Appears in these related concepts: Two Types of Random Variables, Probability Distributions for Discrete Random Variables, and Probability Histograms
 expected value
 Appears in these related concepts: What Does the Law of Averages Say?, Expected Values of Discrete Random Variables, and Expected Return
 experiment
 Appears in these related concepts: Experiments, Descriptive and Correlational Statistics, and Primary Market Research
 finite
 Appears in these related concepts: The Sample Average, Introduction to Sequences, and Summing Terms in an Arithmetic Sequence
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Graphs of Equations as Graphs of Solutions
 independent
 Appears in these related concepts: Fundamentals of Probability, Unions and Intersections, and Party Identification
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 probability
 Appears in these related concepts: Theoretical Probability, Rules of Probability for Mendelian Inheritance, and The Addition Rule
 probability density function
 Appears in these related concepts: Probability, Continuous Sampling Distributions, and Philosophical Implications
 probability theory
 Appears in these related concepts: Chance Processes, Complementary Events, and Independence
 regression
 Appears in these related concepts: Making a Box Model, Email data, and Coefficient of Determination
 residual
 Appears in these related concepts: Plotting the Residuals, Models with Both Quantitative and Qualitative Variables, and Degrees of Freedom
 residuals
 Appears in these related concepts: Two Regression Lines, Inferences of Correlation and Regression, and Midterm elections and unemployment
 sample
 Appears in these related concepts: Identifying Product Benefits, Surveys, and Basic Inferential Statistics
 sample mean
 Appears in these related concepts: Which Standard Deviation (SE)?, Basic properties of point estimates, and Introduction to confidence intervals
 unbiased
 Appears in these related concepts: Samples, Using Impartial Language, and Standard Error
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
 variance
 Appears in these related concepts: Testing a Single Variance, Variance, and Variance Estimates
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