What Is Expected Value? (Definition)
Expected Value Definition
Expected value is the probability-weighted average of all possible outcomes of a random variable. Written E(X) or μ, it represents the theoretical long-run mean you would observe if the experiment were repeated many times under identical conditions. The formula is E(X) = Σ x·P(x): multiply each outcome by its probability, then add all products.
The plain-English translation: multiply each possible outcome by the probability that outcome occurs, then add all those products together. The result is a single number — the "center of gravity" of the entire probability distribution.
Consider a fair six-sided die. The outcomes are 1 through 6, each with probability 1/6. Multiplying and summing: E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 21/6 = 3.5. Roll that die 10,000 times and record every result; the average of all 10,000 numbers will be extremely close to 3.5.
Expected Value Symbol and Notation: E(X), μ, E[X], ⟨X⟩
Expected value is written using four equivalent notations. All four mean the same thing — the probability-weighted average of all outcomes of random variable X:
When you see expressions like E(2X + 3) or E[X + Y], you apply the rules of expectation below. The letter E in E(X) stands for "expectation" — the historical term dating to Christiaan Huygens' 1657 treatise on probability, De ratiociniis in ludo aleæ. The notation E(X) was standardised by W.A. Whitworth in 1901 and has been universal in English-language mathematics since.
- Notation: E(X), E[X], μ (mu), or ⟨X⟩ — all identical in meaning
- Formula (discrete): E(X) = Σ x·P(x) — sum of (outcome × probability)
- Formula (continuous): E(X) = ∫ x · f(x) dx — integral replaces summation
- Probabilities must sum to 1: Σ P(x) = 1 — this is non-negotiable
- Positive EV: The activity gains value on average over repeated trials
- Negative EV: The activity loses value on average — stay away or price it accordingly
- EV = 0: A "fair game" — neither side has a systematic advantage
- E(constant) = constant: E(c) = c — a fixed number has no randomness
The Expected Value Formula Explained
Discrete Random Variables: E(X) = Σ x·P(x)
E(X) = expected value (also written μ)Σ = "sum of" — add every termx = a specific outcome valueP(x) = probability that outcome x occursThe summation symbol Σ (sigma) tells you to repeat the multiplication for every possible value of x and then add the results. If a random variable can take five different values, you get five terms that you add together. If it can take 100 values, you get 100 terms. The formula scales to any number of outcomes.
Continuous Random Variables: E(X) = ∫ x · f(x) dx
∫ = integral (continuous analogue of Σ)f(x) = probability density function at xdx = infinitesimally small x intervalThe expected value integral E(X) = ∫−∞+∞ x · f(x) dx is the continuous analogue of the discrete sum. Where discrete EV uses a probability mass function P(x), continuous EV uses a probability density function f(x). The integral replaces the summation Σ, and the calculus machinery handles the infinite range of possible values. Common continuous distributions and their expected values appear in the distribution table below.
Before computing any expected value, verify that Σ P(x) = 1.0. If your probabilities add to 0.9 or 1.1, the distribution is invalid and your EV calculation will be wrong. This is the single most common error on statistics exams and in real business models.
Expected Value Across Different Contexts
| Context | Discrete Formula | What x Represents | What P(x) Represents |
|---|---|---|---|
| Statistics / Probability Theory | E(X) = Σ x·P(x) | Numeric outcome of a random variable | Probability mass at that outcome |
| Finance / Investment Analysis | EV = Σ Rᵢ·Pᵢ | Return or payoff in dollars | Probability that scenario i occurs |
| Games of Chance / Sports Betting | EV = (P·Win) − (1−P)·Stake | Net gain/loss per play (in dollars) | Probability of winning |
| Insurance / Actuarial Science | E(Loss) = Σ Lᵢ·Pᵢ | Claim amount (in dollars) | Probability that claim occurs |
How to Calculate Expected Value: 4 Steps
List All Outcomes
Write down every value the random variable X can take. Do not skip or combine outcomes — each unique payoff gets its own row.
Assign Probabilities
Record the probability for each outcome. Check that Σ P(x) = 1 before proceeding. This is non-negotiable.
Multiply Each Row
For each outcome, compute the product x × P(x). This gives you the weighted contribution of that outcome to the total average.
Sum All Products
Add every x·P(x) value from Step 3. The total is the expected value. Label it E(X) or μ.
Worked Examples
Example 1 — The Classic: Expected Value of a Die Roll
What is the expected value of one roll of a fair six-sided die?
| Outcome (x) | Probability P(x) | Product x · P(x) |
|---|---|---|
| 1 | 1/6 ≈ 0.1667 | 1 × 0.1667 = 0.1667 |
| 2 | 1/6 ≈ 0.1667 | 2 × 0.1667 = 0.3333 |
| 3 | 1/6 ≈ 0.1667 | 3 × 0.1667 = 0.5000 |
| 4 | 1/6 ≈ 0.1667 | 4 × 0.1667 = 0.6667 |
| 5 | 1/6 ≈ 0.1667 | 5 × 0.1667 = 0.8333 |
| 6 | 1/6 ≈ 0.1667 | 6 × 0.1667 = 1.0000 |
| TOTAL | 6/6 = 1.0000 | E(X) = 3.5000 |
✓ E(X) = 3.5. This is a theoretical average — you can never roll a 3.5 on a single turn. But if you roll 10,000 times, the average of all your results will converge to 3.5.
Example 2 — Coin Flip: Expected Heads in Three Tosses
You flip a fair coin three times. What is the expected number of heads?
| Heads (x) | Ways (combinations) | Probability P(x) | x · P(x) |
|---|---|---|---|
| 0 | TTT → 1 way | 1/8 = 0.125 | 0 × 0.125 = 0.000 |
| 1 | HTT, THT, TTH → 3 ways | 3/8 = 0.375 | 1 × 0.375 = 0.375 |
| 2 | HHT, HTH, THH → 3 ways | 3/8 = 0.375 | 2 × 0.375 = 0.750 |
| 3 | HHH → 1 way | 1/8 = 0.125 | 3 × 0.125 = 0.375 |
| TOTAL | 8 outcomes | 1.000 | E(X) = 1.500 |
✓ E(X) = 1.5 heads per 3-flip session. Confirms the binomial shortcut: E(X) = n·p = 3 × 0.5 = 1.5. See probability trees for a visual breakdown of all 8 outcomes.
Example 3 — Gambling: American Roulette & Betting EV Formula
You bet $1 on a single number in American roulette. What is your expected value?
| Outcome | Net Gain/Loss (x) | Probability P(x) | x · P(x) |
|---|---|---|---|
| Win (your number hits) | +$35 | 1/38 ≈ 0.02632 | +$35 × 0.02632 = +$0.9211 |
| Lose (any other number) | −$1 | 37/38 ≈ 0.97368 | −$1 × 0.97368 = −$0.9737 |
| TOTAL | 1.0000 | E(X) = −$0.0526 |
✓ E(X) = −$0.0526 per $1 wagered. The house edge: for every dollar bet, the player loses 5.26 cents on average over time.
EV = P · (Decimal Odds − 1) − (1 − P)
For sports betting with decimal odds, the expected value formula is: EV = P × (d − 1) − (1 − P), where P is your estimated probability of winning and d is the decimal odds. If the bookmaker offers d = 2.50 (even money at 40% implied probability) but you believe the true win probability is 45%: EV = 0.45 × (2.50 − 1) − (1 − 0.45) = 0.675 − 0.55 = +0.125. A positive EV bet means value in your favour over the long run. See probability rules for the foundations behind this calculation.
Example 4 — Business: Product Launch Decision
Should an entrepreneur launch a new product? Three market scenarios are possible.
| Scenario | Payoff (x) | Probability P(x) | x · P(x) |
|---|---|---|---|
| Strong demand | +$200,000 | 0.25 | +$50,000 |
| Moderate demand | +$20,000 | 0.45 | +$9,000 |
| Weak demand | −$80,000 | 0.30 | −$24,000 |
| TOTAL | 1.00 | E(X) = +$35,000 |
✓ E(X) = +$35,000. Positive EV — the launch is mathematically favorable when assessed over repeated similar decisions. Note: EV alone doesn't capture risk — the 30% chance of losing $80,000 may matter depending on the entrepreneur's risk tolerance. See variance for how to quantify that spread.
Interactive Expected Value Calculator — E(X) = Σ x·P(x)
Use this free expected value calculator to compute E(X) = Σ x·P(x) for any discrete probability distribution. Enter your outcomes and probabilities below — the calculator verifies that probabilities sum to 1 and shows the full step-by-step calculation. Add up to 10 outcome rows. For a dedicated calculator page, visit our standalone expected value calculator.
Expected Value Calculator — Discrete Random Variable
Rules of Expected Value (All Six Properties)
Rules of Expectation: Core Properties
The rules of expected value tell you how E(X) behaves under arithmetic operations. Mastering these rules lets you compute expected values for complex expressions without rebuilding a full distribution table each time.
Linearity of Expectation
The expected value of a sum always equals the sum of expected values — even when X and Y are not independent. This is one of the most powerful and universally applicable properties in probability theory.
Linear Transformation
Scaling a variable by constant a scales its expected value by a. Shifting by constant b shifts the expected value by b. If E(X) = 3 and Y = 2X + 5, then E(Y) = 2(3) + 5 = 11.
Expected Value of a Constant
If c is a constant (not random), its expected value is simply c itself. E(5) = 5. E(−3) = −3. A constant has no uncertainty — it always takes that exact value, so the probability-weighted average equals the value itself. This rule is used in Rule 2 above: the +b term in E(aX+b) = a·E(X)+b comes from E(b) = b.
Expected Value of a Product
When X and Y are independent, the expected value of their product equals the product of their expected values. This does not hold when variables are dependent. This asymmetry is why E(X) is called linear but not multiplicative: addition always works (Rule 1), multiplication only works under independence.
Scaling Rule
Multiplying a random variable by a constant scales the expected value by the same constant. If E(X) = 4 and Y = 3X, then E(Y) = 3 × 4 = 12. This follows directly from Rule 2 with b = 0.
Iterated Expectation
The expected value of the conditional expectation of X given Y equals the unconditional expected value of X. This is the Law of Total Expectation — fundamental in Bayesian statistics and conditional probability.
Positive vs Negative Expected Value
Can Expected Value Be Negative?
| E(X) Sign | Meaning | Real-World Example |
|---|---|---|
| E(X) > 0 (Positive) | Average gain over many trials; activity has long-run upside | Positive-EV poker play, index fund investing, insurance underwriting |
| E(X) < 0 (Negative) | Average loss over many trials; activity has long-run downside | Casino slots (player side), lottery tickets, payday loans (borrower side) |
| E(X) = 0 (Fair Game) | No systematic advantage for either party over repeated play | Theoretical fair coin-flip bet at even odds; zero-sum trading with no fees |
Expected Value and the Law of Large Numbers
You cannot roll the expected value of 3.5 on a single die throw. The expected value is not a prediction for any individual event — it is the mathematical limit of what the sample mean approaches as the number of trials grows without bound.
As n → ∞, the sample mean x̄ converges to E(X) in probability.
In plain English: the more trials you run, the closer your observed average gets to the theoretical expected value. This result underpins all of inferential statistics and explains why casinos and insurance companies make reliable profits even when individual outcomes vary wildly. For a deep dive, see our Law of Large Numbers guide.
The St. Petersburg Paradox — When EV Breaks Down
Flip a fair coin until tails appears. You win $2n where n is the flip number. The expected value is infinite: E(X) = Σ 2k·(1/2)k = 1 + 1 + 1 + … = ∞. Yet rational people would pay only a few dollars to play. This paradox — published in 1738 by Daniel Bernoulli in the Commentaries of the Imperial Academy of Science of Saint Petersburg — shows that E(X) alone is insufficient when outcomes are extreme. Real decisions require variance and utility alongside expected value. The paradox remains unresolved and is one of the founding puzzles of decision theory.
Discrete vs Continuous Random Variables
| Feature | Discrete Random Variable | Continuous Random Variable |
|---|---|---|
| Possible values | Countable list: 0, 1, 2, 3, … | Infinite range: any value in [a, b] |
| Probability function | PMF: P(X = x) | PDF: f(x) |
| E(X) formula | E(X) = Σ x · P(x) | E(X) = ∫ x · f(x) dx |
| Examples | Number of heads, die face, number of customers | Height, weight, time until failure, stock price change |
| Tools needed | Arithmetic — multiplication and addition | Calculus — integration |
Expected Value of Common Probability Distributions
Each named probability distribution has a closed-form expected value formula. The table below summarises the most common distributions, their E(X) formulas, and what the parameters mean. For worked examples with each distribution, see the dedicated guides linked in Related Topics.
| Distribution | E(X) Formula | Parameters | Use Case |
|---|---|---|---|
| Bernoulli | E(X) = p | p = success probability | Single yes/no trial (coin flip) |
| Binomial B(n,p) | E(X) = n·p | n = trials, p = P(success) | Count of successes in n trials |
| Geometric | E(X) = 1/p | p = P(success per trial) | Trials until first success |
| Poisson (λ) | E(X) = λ | λ = average rate | Count of rare events in fixed interval |
| Uniform [a, b] | E(X) = (a+b)/2 | a = min, b = max | All values equally likely in [a,b] |
| Exponential (λ) | E(X) = 1/λ | λ = rate parameter | Time between events in a Poisson process |
| Normal N(μ,σ²) | E(X) = μ | μ = mean, σ = SD | Heights, measurements, errors |
| Negative Binomial | E(X) = r/p | r = successes needed, p = P(success) | Trials until r successes |
The Binomial EV E(X) = n·p is the most frequently tested formula: it follows directly from applying linearity of expectation to n independent Bernoulli(p) trials. See our Binomial Distribution guide and the related Poisson Distribution for full derivations.
Expected Value vs Mean vs Variance
Is Expected Value the Same as Mean?
Yes — expected value (E(X)) and population mean (μ) refer to the same concept when applied to a probability distribution. E(X) is the notation used in probability theory; μ is the standard notation in statistics. The sample mean (x̄) is different — it is computed from actual data and converges to E(X) by the Law of Large Numbers. This distinction is also explored in our dedicated mean vs expected value guide and the broader discussion of the weighted mean — which E(X) is a probability-weighted instance of.
E(X) = μ
The center of the distribution. Measures where outcomes cluster on average.
Var(X) = σ²
The average squared deviation from the mean. Measures how spread out outcomes are. Shortcut: E(X²) − [E(X)]².
SD(X) = σ
The square root of variance — restores the original units of X. Easier to interpret because σ is in the same units as the outcomes.
Entity & Formula Glossary
| Term | Notation / Formula | Plain-English Definition |
|---|---|---|
| Expected Value | E(X), E[X], μ, ⟨X⟩ | The probability-weighted average of all possible outcomes of a random variable; the long-run mean. |
| Discrete EV Formula | E(X) = Σ x·P(x) | Sum of every outcome multiplied by its probability. Applies when X takes a countable set of values. |
| Continuous EV Formula | E(X) = ∫ x·f(x) dx | Integral of outcome times probability density. Applies when X can take any value in a continuous range. |
| Summation Symbol | Σ (sigma) | "Add up all terms." Σ x·P(x) means: compute x·P(x) for each possible x, then add all those products. |
| Outcome | x (or xᵢ) | A specific numeric value that the random variable X can take in a single trial. |
| Probability of Outcome | P(x) or P(X = x) | The likelihood (between 0 and 1) that outcome x occurs in a single trial. |
| Linearity of Expectation | E(X+Y) = E(X)+E(Y) | Expected values of sums are additive — holds for any variables, whether independent or not. |
| Linear Transformation Rule | E(aX+b) = a·E(X)+b | Scaling and shifting a variable scales and shifts its expected value by the same amounts. |
| Constant Rule | E(c) = c | The expected value of a constant is the constant itself — no randomness means the average is the value. |
| Product Rule (independent) | E(X·Y) = E(X)·E(Y) | Expected value of a product equals the product of expected values — only when X and Y are independent. |
| Variance | Var(X) = E[(X−μ)²] | The expected squared deviation from the mean. Shortcut: E(X²) − [E(X)]². |
| Negative Expected Value | E(X) < 0 | On average, repeated participation results in a net loss. Example: casino games (player side). |
| Law of Large Numbers | x̄ₙ → μ as n → ∞ | The sample mean of n trials converges to the expected value as n grows. |
Expected Value Formula Cheat Sheet & Rules Reference
| Formula Name | Notation | When to Use | Plain-English Meaning |
|---|---|---|---|
| Discrete Expected Value | E(X) = Σ x·P(x) | Countable outcomes (die, coins, counts) | Multiply each outcome by its probability; add all products. |
| Continuous Expected Value | E(X) = ∫ x·f(x) dx | Continuous range (heights, times, prices) | Integrate outcome × density over all values. |
| Linearity Rule | E(X+Y) = E(X)+E(Y) | Any two variables — always | Break complex EVs into simpler parts and add results. |
| Scaling Rule | E(aX+b) = a·E(X)+b | Transformed or rescaled variables | Scale the EV, then shift it — same as transforming the center. |
| Constant Rule | E(c) = c | Fixed (non-random) values | The average of something fixed is just that thing. |
| Product Rule (independent) | E(X·Y) = E(X)·E(Y) | Independent random variables only | Multiply EVs only when the variables do not influence each other. |
| Variance Shortcut | Var(X) = E(X²) − [E(X)]² | Computing spread after EV is known | Subtract squared mean from mean of squared outcomes. |
| Binomial EV | E(X) = n·p | n independent trials, each with P(success)=p | Number of trials × probability of success per trial. |
| Geometric EV | E(X) = 1/p | Waiting for first success | Expected number of trials until the first success. |
| Poisson EV | E(X) = λ | Count of rare events in fixed interval | Rate parameter λ is both the mean and variance. |
| Betting EV (decimal odds) | EV = P·(d−1) − (1−P) | Sports betting with decimal odds d | Your edge per unit staked; positive = value bet. |
Quick-Answer Reference Block
What Is Expected Value? (Definition for Featured Snippet)
What Is E(X) in Probability and Statistics?
E(X) means "expected value of X" — the probability-weighted average of all outcomes.
E(X) is the most common notation for expected value. It is also written E[X], μ (mu), or ⟨X⟩. In probability theory, E(X) = Σ x·P(x) for discrete variables; in calculus-based statistics, E(X) = ∫ x·f(x) dx for continuous variables. The letter E stands for "expectation." When you see E(2X + 3), you apply the linear transformation rule: E(2X + 3) = 2·E(X) + 3.
Common Misconceptions and Pitfalls
Expected value applies to repeated trials, not individual events. A single trial can produce any outcome, including ones far from E(X). Misapplying EV to one-off, irreversible decisions without considering variance and risk tolerance is a systematic error in decision analysis.
If Σ P(x) ≠ 1.0, your probability distribution is invalid. E(X) calculated from an invalid distribution is mathematically meaningless. Always verify the sum before computing. See probability rules for the axioms that govern valid distributions.
Two investments can have identical positive expected values but wildly different risk profiles. Same EV; completely different risk. In real decisions, variance matters alongside expected value.
E(X²) is the expected value of X-squared — a different computation from squaring E(X). These are not equal unless X is a constant. The correct variance formula is Var(X) = E(X²) − [E(X)]².
E(X·Y) = E(X)·E(Y) is true only when X and Y are independent. For correlated variables (e.g., stock returns that move together), you must use the covariance formula: E(X·Y) = E(X)·E(Y) + Cov(X,Y). Skipping this correction leads to systematically wrong risk estimates in finance.
Related Statistical Concepts
Basic Probability
Every P(x) in the EV formula is a probability. Mastering the rules of probability is prerequisite to computing meaningful expected values.
Probability Rules
The axioms that guarantee ΣP(x) = 1 — the critical prerequisite for every valid expected value calculation.
Random Variables
Expected value is a property of a random variable's distribution. Understanding discrete vs continuous variables determines which EV formula to apply.
Binomial Distribution
For n Bernoulli trials with success probability p, E(X) = np — derived directly from the general formula.
Poisson Distribution
For a Poisson variable with rate λ, E(X) = λ — the rate parameter is both mean and variance.
Variance & Standard Deviation
Once E(X) is known, Var(X) = E(X²) − [E(X)]² quantifies how spread out outcomes are.
Weighted Mean
Expected value is a probability-weighted mean — understanding weighted averages gives intuition for the E(X) formula.
Conditional Probability
Conditional expectation E(X|Y) extends EV to cases where you already know something about the outcome.
For authoritative external references: OpenStax Introductory Statistics — Expected Value, Khan Academy — Expected Value, and Wolfram MathWorld — Expectation Value.