Expected Value Calculator [2025]
Calculate the expected value, variance, and standard deviation of probability distributions with visual aids.
Expected Value Calculator
Calculate the expected value, variance, and standard deviation of a probability distribution
About Expected Value
The expected value (or mean) of a random variable is the long-run average value of repetitions of the experiment. It represents the weighted average of all possible values a random variable can take.
For a discrete random variable X with possible values x₁, x₂, ..., xₙ and corresponding probabilities p₁, p₂, ..., pₙ:
E[X] = x₁p₁ + x₂p₂ + ... + xₙpₙ = Σ(xᵢpᵢ)
The variance measures the spread of values around the expected value, and the standard deviation is the square root of the variance.
Key Features
- Calculate the expected value (mean), variance, and standard deviation of custom probability distributions.
- Built-in support for common discrete distributions: Binomial, Poisson, Uniform, Geometric, and a Normal approximation.
- Interactive bar chart visualization to help understand probability distributions.
- Detailed step-by-step solutions showing the mathematical process.
- Easy input for custom probability distributions with automatic validation.
- Intuitive parameter controls for predefined distributions.
- Copy results directly to clipboard for use in other applications.
- Load example data to quickly understand tool functionality.
Why Use the Expected Value Calculator?
Expected value is a fundamental concept in probability theory and statistics, representing the long-run average outcome of a random experiment. Our calculator makes computing expected values, variances, and standard deviations simple and accessible, whether you're a student learning statistics, a professional analyzing data, or just curious about probability.
With support for both custom probability distributions and common predefined distributions, this tool is versatile enough for a wide range of applications. The interactive visualizations and step-by-step calculations help develop intuition about probability concepts while providing accurate results for your statistical needs.
Understanding Expected Value
The expected value (or mean) of a random variable represents the average of all possible values, each weighted by its probability of occurrence. In simpler terms, if you repeated an experiment many times, the average of all outcomes would approach the expected value.
For a discrete random variable:
E[X] = x₁·p₁ + x₂·p₂ + ... + xₙ·pₙ = Σ(xᵢ·pᵢ)
Where x₁, x₂, ..., xₙ are the possible values and p₁, p₂, ..., pₙ are their respective probabilities.
Variance and Standard Deviation:
Variance measures the spread or dispersion of a probability distribution around its expected value:
Var(X) = E[(X - E[X])²] = Σ((xᵢ - E[X])² · pᵢ)
The standard deviation is simply the square root of the variance:
σ = √Var(X)
Understanding these concepts is crucial in many fields, including finance (expected returns), insurance (risk assessment), game theory (expected payoffs), and many scientific disciplines.
Common Probability Distributions
Binomial Distribution
Models the number of successes in a fixed number of independent trials, each with the same probability of success. Used for scenarios like coin flips, yes/no surveys, or pass/fail tests.
E[X] = n·p, Var(X) = n·p·(1-p)
Where n is the number of trials and p is the probability of success.
Poisson Distribution
Models the number of events occurring in a fixed interval of time or space, assuming events occur independently. Used for modeling rare events like accidents, arrivals, or defects.
E[X] = λ, Var(X) = λ
Where λ is the average rate of occurrence.
Uniform Distribution
Models a random variable with equal probability across all values in a fixed range. Used for scenarios like rolling a fair die or random selection from a range.
E[X] = (a+b)/2, Var(X) = ((b-a+1)²-1)/12
Where a is the minimum value and b is the maximum value.
Geometric Distribution
Models the number of trials needed to get the first success in a series of independent trials. Used for scenarios like the number of coin flips until the first heads.
E[X] = 1/p, Var(X) = (1-p)/p²
Where p is the probability of success in a single trial.
Normal Distribution
A continuous probability distribution that is symmetric about the mean. Used extensively in statistics due to the Central Limit Theorem and for modeling natural phenomena.
E[X] = μ, Var(X) = σ²
Where μ is the mean and σ is the standard deviation.
Frequently Asked Questions
Why do probabilities need to sum to 1?
Probabilities must sum to 1 because they represent all possible outcomes of a random experiment. A total probability of 1 (or 100%) means that all possibilities have been accounted for in your distribution.
What is the difference between expected value and average?
The expected value is a theoretical concept that represents the long-run average outcome if an experiment were repeated many times. The average (or sample mean) is calculated from observed data. As the number of observations increases, the average should approach the expected value.
Can the expected value be a value that is impossible to actually observe?
Yes! For example, the expected value when rolling a fair six-sided die is 3.5, even though it's impossible to actually roll a 3.5. The expected value represents a weighted average, not necessarily a possible outcome.
How is standard deviation related to variance?
Standard deviation is simply the square root of variance. While variance measures the average squared deviation from the expected value, standard deviation brings this measure back to the original units, making it more interpretable.
When should I use which distribution?
Choose the distribution that best models your scenario:
- Binomial: Fixed number of independent yes/no trials
- Poisson: Random events occurring at a constant rate
- Uniform: Equal likelihood across a range of values
- Geometric: Trials until first success
- Normal: Many natural phenomena and aggregations of random variables
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