Mean of Probability Distribution Calculator
The Mean of Probability Distribution Calculator computes the expected value of a random variable. It is used in statistics, finance, and data science to predict average outcomes in scenarios involving uncertainty, such as risk analysis, decision-making, and probability modeling.
Calculator
Formula
The mean (μ) of a probability distribution is calculated as: μ = Σ [xi * P(xi)], where xi are possible values and P(xi) their corresponding probabilities.
How to Use
Enter numeric values separated by commas in the "Values" field and their corresponding probabilities in the "Probabilities" field. Ensure probabilities sum to 1. Click "Calculate" to get the mean. Use "Clear" to reset inputs.
Derivation Process
The mean is derived by multiplying each possible outcome by its probability of occurrence and summing all products. This weighted average represents the expected outcome over many trials, rooted in the law of large numbers and probability axioms.
FAQs
1. Why is the mean important in probability distributions?
The mean provides the expected average outcome, helping predict long-term behavior in random experiments. It's crucial for decision-making under uncertainty in fields like finance and engineering.
2. Can the mean differ from individual values?
Yes, the mean is a weighted average and may not match any specific outcome. It balances high and low values based on their probabilities.
3. What if probabilities don't sum to 1?
The calculator alerts an error. Valid probability distributions require ΣP(x) = 1. Normalize probabilities if needed.
4. How many values can I input?
The calculator handles any number of values, but ensure equal entries in both fields. Large datasets may need spreadsheet tools.
5. Is this calculator for discrete distributions only?
Yes, this tool is for discrete variables. Continuous distributions require integration, handled differently.
6. What if I enter non-numeric values?
The calculator shows an error. Ensure all inputs are numbers separated by commas.
7. Can I use percentages for probabilities?
No, enter probabilities as decimals (e.g., 0.25 for 25%). The tool validates decimal format.
8. How is this different from sample mean?
This calculates theoretical expected value, not empirical average. Sample mean uses observed data.
9. What's the relation to variance?
Variance measures spread around the mean. Calculate it using E(x²) - [E(x)]² after finding the mean.
10. Is my data secure?
Yes, all calculations occur client-side. No data is stored or transmitted.