Expected Value: Average Value Over Many Observations

August 25, 2024 4 min read Mathematics Statistics Expected Value Statistics Random Variable Probability Expectation

The expected value represents the average value that a random variable would yield if observed many times, also known as the expectation.

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The expected value (EV), also known as expectation, mathematical expectation, or mean, is a fundamental concept in probability and statistics. It provides the average outcome of a random variable if the experiment were to be repeated an infinite number of times. Mathematically, it is the weighted average of all possible values of the random variable, where each value is weighted by its probability of occurrence.

Mathematical Definition

For a discrete random variable $X$ with possible values $x_1, x_2, …, x_n$ and corresponding probabilities $P(X = x_1), P(X = x_2), …, P(X = x_n)$, the expected value $E(X)$ is given by:

E(X) = \sum_{i=1}^{n} x_i P(X = x_i)

For a continuous random variable $X$ with a probability density function $f(x)$, the expected value is defined as:

E(X) = \int_{-\infty}^{\infty} x f(x) \, dx

Types of Random Variables

Discrete Random Variables

These are random variables that can take on a finite or countably infinite set of distinct values. For example, the outcome of rolling a die is a discrete random variable.

Continuous Random Variables

These random variables can take on any value within a given range. An example includes the height of individuals within a population.

Special Considerations

Law of Large Numbers

The Law of Large Numbers states that as the number of trials increases, the sample average converges to the expected value. Therefore, with a sufficiently large number of observations, the average of the observed values will approximate the expected value.

Variance and Standard Deviation

While the expected value provides a measure of central tendency, it does not convey information about the dispersion of the data. This is where variance and standard deviation come into play, measuring how much values differ from the expected value.

Examples

Rolling a Fair Six-Sided Die:
- Possible values: 1, 2, 3, 4, 5, 6
- Probability for each value: $1/6$
- Expected value:
$$ E(X) = \sum_{i=1}^{6} x_i P(X = x_i) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = 3.5 $$
Continuous Distribution (Uniform Distribution $U(a, b)$):
- For a uniform distribution between $a$ and $b$, the expected value is:
$$ E(X) = \frac{a + b}{2} $$

For $U(0, 1)$, the expected value is:

$$ E(X) = \frac{0 + 1}{2} = 0.5 $$

Historical Context

The concept of expected value has its origins in gambling problems and was introduced by Christiaan Huygens in the 17th century. It was further developed by Pierre-Simon Laplace and Carl Friedrich Gauss, becoming a cornerstone of probability theory and statistics.

Applicability

Finance and Economics

In finance, the expected value is used in decision-making and risk management, including investments and insurance. For instance, the expected return of an investment portfolio is crucial for financial planning.

Science and Engineering

Expected values are used in scientific experiments to predict outcomes and analyze data, and in engineering to assess reliability and system performance.

Surveys and social experiments often rely on expected values to draw inferences about populations or behavior patterns.

Comparisons

Mean vs. Median: While the mean (or expected value) is the arithmetic average, the median is the middle value when data is sorted. In skewed distributions, the mean and median can differ significantly.
Expected Value vs. Mode: The mode is the value that appears most frequently in a data set, whereas the expected value is a probabilistic average.

Random Variable: A variable whose values depend on outcomes of a random phenomenon.
Probability Distribution: A function that describes the likelihood of various outcomes.
Variance: Measures the dispersion of a set of values relative to their mean.
Standard Deviation: The square root of the variance, providing a measure of dispersion.

FAQs

What is the importance of expected value in decision-making?

The expected value is crucial in decision-making processes where outcomes under uncertainty are considered. It provides a rational basis for making choices that maximize long-term average returns.

How is expected value different from the probability?

Expected value is a weighted average of all possible outcomes, whereas probability measures the likelihood of a single specific outcome.

Can the expected value be negative?

Yes, the expected value can be negative if the weighted average of the possible values results in a negative number. This is common in scenarios involving losses or costs.

References

Huygens, Christiaan. “De ratiociniis in ludo aleae” (1657).
Laplace, Pierre-Simon. “Théorie analytique des probabilités” (1812).
Ross, Sheldon M. “A First Course in Probability” (10th Edition, 2018).

Summary

The expected value is a core concept in probability and statistics, representing the average outcome of a random variable over many observations. It serves as a foundational tool in various fields, including finance, science, engineering, and social sciences, helping to predict outcomes and guide decision-making processes. Understanding the expected value allows for deeper insights into the behavior of random variables and the distributions they follow.