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Expected Value Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

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)

1.
Value x
Prob p
2.
Value x
Prob p
3.
Value x
Prob p
Distribution Stats

Expected Value E(X)

21

Total Probability: 1

Expected Value E(X)21
Variance Var(X)49
Standard Dev SD(X)7

Summation formula:

E(X) = (10 × 0.2) + (20 × 0.5) + (30 × 0.3) = 2 + 10 + 9 = 21
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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.