\[ \text{Var} = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]
Where:
Beta variance refers to the variance of a beta distribution, which is a continuous probability distribution defined on the interval [0, 1]. The beta distribution is parameterized by two positive shape parameters, alpha (\( \alpha \)) and beta (\( \beta \)), which determine the shape of the distribution. The variance of the beta distribution provides a measure of the spread or dispersion of the distribution around its mean. It is an important characteristic in statistics and probability theory, especially in Bayesian statistics and various applications involving proportions and probabilities.
Alpha (\( \alpha \)): 2
Beta (\( \beta \)): 5
Variance: \[
\frac{2 \times 5}{(2 + 5)^2 \times (2 + 5 + 1)} = \frac{10}{49 \times 8} \approx 0.02551
\]
Alpha (\( \alpha \)): 10
Beta (\( \beta \)): 10
Variance: \[
\frac{10 \times 10}{(10 + 10)^2 \times (10 + 10 + 1)} = \frac{100}{400 \times 21} \approx 0.01190
\]
