Calculadora de Distribuição Beta (E[X])

The Beta Distribution Calculator computes the mean (expected value E[X]) and variance of a random variable following the Beta distribution. The Beta distribution is defined on the interval [0, 1] and is parameterized by two positive parameters α (alpha) and β (beta). The formula for the mean is E[X] = α / (α + β), and the variance is Var = αβ / ((α+β)²(α+β+1)). This tool is useful for professionals working with probabilistic models, such as in risk analysis, Bayesian statistics, and Monte Carlo simulations.

To use the calculator, enter the values of α and β (both greater than zero). The result displays the mean and variance of the distribution. The mean represents the expected central value, while the variance measures the spread of data around the mean. For example, if α=2 and β=5, the mean is 2/(2+5) ≈ 0.2857, indicating that the variable tends to values closer to 0. The variance helps understand the associated uncertainty.

When to use this calculator? It is ideal for modeling probabilities and proportions, such as conversion rates in marketing, probability of success in experiments, or opinions in surveys. It is also used in industrial processes for quality control and in finance to model percentage returns. The Beta distribution is flexible: with equal α and β, it is symmetric; with larger α, it tends to 1; with larger β, it tends to 0.

Caution: ensure α and β are positive numbers. Negative or zero values are invalid. Interpreting results depends on context; the mean alone does not capture all uncertainty. For very small α or β, the distribution may be highly concentrated at the extremes. Always verify that the Beta distribution is appropriate for your data, especially if the phenomenon is not limited to the [0,1] interval.