Expected Value

Statistics
definition

Also known as: mean, E(X)

Grade 9-12

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The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions. Basis for decision-making under uncertainty, insurance, gambling.

Definition

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

💡 Intuition

Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.

🎯 Core Idea

E[X] = \sum x_i P(x_i): multiply each outcome by its probability and sum. Expected value is linear: E[aX + b] = aE[X] + b.

Example

Fair die: E(X) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5 You never roll 3.5, but it's the average.

Formula

E(X) = \sum[x \times P(x)]

Notation

E(X) or \mu_X denotes the expected value of random variable X

🌟 Why It Matters

Basis for decision-making under uncertainty, insurance, gambling.

💭 Hint When Stuck

Make a two-column table: each possible outcome and its probability. Multiply across each row, then add all the products.

Formal View

E(X) = \sum_{i} x_i \, P(X = x_i) (discrete); E(X) = \int_{-\infty}^{\infty} x \, f(x) \, dx (continuous)

Related Concepts

ProbabilityMeanVariance

🚧 Common Stuck Point

A game is 'fair' when expected value = 0 (break even long-term).

⚠️ Common Mistakes

  • Forgetting to weight each outcome by its probability — simply averaging the possible values without accounting for their likelihood
  • Expecting the expected value to occur on a single trial — E(X) = 3.5 for a die does not mean you will ever roll 3.5
  • Adding probabilities instead of multiplying each outcome by its probability before summing

Frequently Asked Questions

What is Expected Value in Math?

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

Why is Expected Value important?

Basis for decision-making under uncertainty, insurance, gambling.

What do students usually get wrong about Expected Value?

A game is 'fair' when expected value = 0 (break even long-term).

What should I learn before Expected Value?

Before studying Expected Value, you should understand: probability, mean.

Prerequisites

probabilitymean

Next Steps

variance

How Expected Value Connects to Other Ideas

To understand expected value, you should first be comfortable with probability and mean. Once you have a solid grasp of expected value, you can move on to variance.

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Visualization

Static

Visual representation of Expected Value