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Expected Value Calculator
Last updated: June 19, 2026
An expected value calculator computes the theoretical mean (expected value) of a discrete random variable. By summing the product of each possible outcome and its respective probability, the tool evaluates the long-term average outcome of a random process. It also validates that all probabilities sum to exactly 1.0 and reports the distribution's variance and standard deviation.
Calculate the expected value, variance, and standard deviation of a discrete random variable distribution.
Quick Answer
Calculate the expected value E(X) of a discrete probability distribution. Enter values and their probabilities to find the mean, variance, and standard deviation.
Probability Distribution Rows (x_i, p_i)
Expected Value E(X)
21
Total Probability: 1
Summation formula:
How it works
Expected Value Formula
For a discrete random variable X with possible values x₁, x₂, ..., xₙ and corresponding probabilities p₁, p₂, ..., pₙ, the expected value E(X) is calculated as:
E(X) = ∑ (xᵢ · pᵢ) = x₁p₁ + x₂p₂ + ... + xₙpₙ
Variance and Standard Deviation
In addition to the mean, we can measure the dispersion of the distribution:
- Variance (Var(X)): The expected value of the squared deviation from the mean:Var(X) = ∑ [ pᵢ · (xᵢ − E(X))² ]
- Standard Deviation (SD(X)): The square root of the variance:SD(X) = √Var(X)
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Frequently asked questions
The expected value, E(X) or μ, of a discrete random variable is the probability-weighted average of all possible values. It represents the long-term average outcome if the random experiment is repeated many times.
Multiply each possible value xᵢ by its probability pᵢ, and sum all the products: E(X) = Σ(xᵢ · pᵢ).
For a discrete distribution to be mathematically valid, the set of outcomes must cover 100% of the possible sample space. The sum of all individual probabilities must equal exactly 1.0 (or 100%).
Expected value is the theoretical mean of a probability distribution, representing what would happen over infinite trials. The sample mean is the average of actual observed data in an experiment.
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