Expected Value Calculator

Compute the expected value E[X] from a discrete probability distribution table.

Inputs

Values (comma separated)

Probabilities (comma separated)

Expected Value E[X]

3.500000

Variance Var(X)

2.050000

Standard Deviation

1.431782

Step by step

Values used
Values (comma separated) = 1, 2, 3, 4, 5, 6; Probabilities (comma separated) = 0.1, 0.15, 0.25, 0.25, 0.15, 0.1
Expected Value
E[X] = Σ xᵢ × P(xᵢ)
Variance
Var(X) = Σ P(xᵢ) × (xᵢ − E[X])²
Expected Value E[X]
= 3.500000
Variance Var(X)
= 2.050000
Standard Deviation
= 1.431782

How it works

The expected value E[X] is the long-run average of a random variable. For a discrete distribution, it is the probability-weighted sum of all possible values: E[X] = Σ xᵢ × P(xᵢ).

Formulas

Expected Value

E[X] = Σ xᵢ × P(xᵢ)

xᵢ: Possible value
P(xᵢ): Probability of that value

Variance

Var(X) = Σ P(xᵢ) × (xᵢ − E[X])²

xᵢ: Value
E[X]: Expected value

Frequently Asked Questions

Does the expected value have to be a possible outcome?

No. For example, the expected value of a fair die roll is 3.5, which is not a possible outcome. It represents the long-run average.

What is the relationship between expected value and mean?

For a probability distribution, the expected value IS the mean (μ). For sample data, the sample mean estimates the population expected value.

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