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. Expected value is the basis for decision-making under uncertainty — from insurance pricing and gambling odds to stock portfolio valuation and clinical trial evaluation, it tells you the long-run average payoff of any random process.

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

Expected value is the basis for decision-making under uncertainty — from insurance pricing and gambling odds to stock portfolio valuation and clinical trial evaluation, it tells you the long-run average payoff of any random process.

💭 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.

What is the Expected Value formula?

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

When do you use Expected Value?

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

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