## What is the Expected Value?

The expected value in statistics is the long-run average outcome of a random variable based on its possible outcomes and their respective probabilities. Essentially, if an experiment (like a game of chance) were repeated, the expected value tells us the average result we’d see in the long run. Statisticians denote it as E(X), where E is “expected value,” and X is the random variable.

Understanding the expected value isn’t just a mathematical exercise—it offers a predictive lens through which we can forecast likely outcomes in uncertain scenarios, from financial investments to decision-making in various fields.

Its origins trace back to the 17th century when famed mathematicians Blaise Pascal and Pierre de Fermat corresponded about gambling problems. They sought a rational way to divide stakes in a game interrupted before its conclusion.

The concept of expected value emerged from these early deliberations on fair bets and equitable decisions, providing gamblers, scholars, investors, and decision-makers with a robust tool to predict, strategize, and optimize in the face of uncertainty.

In this post, learn how to find an expected value for different cases and calculate it using formulas for various probability distributions. We’ll work through example calculations for expected values in several contexts.

Learn more about Random Variables: Discrete & Continuous.

## How to Find an Expected Value

Understanding the general process for calculating the expected value for a discrete random variable is the best place to start. A discrete random variable represents specific, distinct values, often arising from countable outcomes like flipping coins, rolling dice, or counting objects.

To find the expected value, multiply each possible value of your discrete variable by its probability and then sum all these products.

The expected value formula for a discrete variable is the following:

Where:

- X is the random variable.
- x
_{i}are the specific values. - i is the index variable from 1 to n, all possible values of the discrete variable.
- n is the number of possible outcomes.
- P(x
_{i}) is the probability of x_{i}.

This method is a fancy version of a weighted average, where each outcome’s weight is its probability. Because the probabilities sum to 1, there is no need to divide. Learn more about Weighted Averages: Formula & Calculation Examples.

Imagine a game show where you have a 0.5 chance to win $100, a 0.4 chance to win $500, and a 0.1 probability to lose $100 (negative because you lose money).

To find the expected value for the game show, we’ll take each outcome (the winnings and loss), multiply it by its probability, and sum them.

E(X) = 100 * 0.5 + 500 * 0.4 – 100 * 0.1 = $240.

Hence, the expected value for that game is $240. As more people play this game, the average outcome will converge on this value according to the law of large numbers.

Having understood the basics, let’s delve into some specific contexts that use variations of the general process I describe above.

### Equiprobable Outcomes

When outcomes are equally likely (like rolling a fair die), the expected value formula is a straightforward average of the results because all the weights (i.e., probabilities) are equal.

For example, when you roll a die, each outcome (1 through 6) has an equal 1/6 chance.

E(X) = 1 + 2 +3 + 4 +5 + 6 / 6 = 3.5

Therefore, the expected value of rolling a die is 3.5. When you roll a die many times, the average will converge on this value.

### Expected Value of a Binomial Distribution

One of the classic applications of an expected value lies within the realm of the binomial distribution. The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials (think “yes-no” scenarios, like flipping a coin). The expected value formula for a binomial distribution is the following:

E(X) = n * p

Where:

- n is the number of trials.
- p is the probability of success on a single trial.

Imagine you’re taking a 10-question multiple-choice quiz where each question has five choices, and you’re guessing on every question. The probability of getting a question right by guessing is 1 out of 5 options or 0.2.

Here:

- n = 10
- p = 0.2

Using the expected value formula for the binomial distribution:

E(X) = 10 * 0.2 = 2

So, if you were to guess randomly on this quiz, you’d expect to answer two questions correctly on average.

### Continuous Probability Distributions

The expected value for a continuous probability distribution is the mean of the random variable. As you take larger random samples from a continuous probability distribution, the sample averages will tend to converge on the expected value thanks to the law of large numbers.

However, calculating the mean for a continuous probability distribution is more complex because continuous probabilities apply to a range rather than a distinct value. Consequently, we need to use integration, which you can think of as summing infinitely tiny products of values and their probabilities.

Fortunately, these solutions reduce to simple expected value formulas. The table below summarizes them for various discrete and continuous distributions.

Learn more about Probability Distributions: Definition & Calculations.

## Expected Value Formulas for Common Probability Distributions

Before we wrap up, here’s a quick cheat sheet on expected value formulas for common probability distributions. Click the links to learn more about each distribution.

Distribution Name |
Expected Value Formula |
Description |

Uniform (a, b) | (a + b) / 2 | Average of the smallest (a) and largest (b) values. |

Binomial (n, p) | np | Number of trials (n) multiplied by the probability of success (p). |

Geometric (p) | 1/p | Reciprocal of the probability of success (how many trials on average until a success). |

Poisson (λ) | λ | Average number of occurrences over a fixed interval or region. |

Exponential (λ) | 1/λ | Reciprocal of the rate parameter (average time between occurrences). |

Normal (μ, σ) | μ | Central location (mean) of the distribution. |

With this guide, you’re well-equipped to make informed predictions and understand the average outcomes of various experiments and scenarios.

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