To prove that the variance of a beta-distributed random variable with parameters $\alpha$ and $\beta$ is given by:

$\sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$

we start by recalling the properties of a beta distribution, specifically the mean and variance formulas for a beta-distributed random variable $X \sim \text{Beta}(\alpha, \beta)$.

Step 1: Mean of a Beta Distribution

The mean $\mathbb{E}[X]$ of a beta distribution with parameters $\alpha$ and $\beta$ is given by:

$\mathbb{E}[X] = \frac{\alpha}{\alpha + \beta}$

Step 2: Variance of a Beta Distribution

The variance $\text{Var}(X)$ of a beta distribution with parameters $\alpha$ and $\beta$ is given by:

[ \text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 ]

Now, we proceed by finding $\mathbb{E}[X^2]$ using the second moment formula.

Step 3: Second Moment $\mathbb{E}[X^2]$

For a beta distribution, the second moment $\mathbb{E}[X^2]$ is known to be:

$\mathbb{E}[X^2] = \frac{\alpha (\alpha + 1)}{(\alpha + \beta)(\alpha + \beta + 1)}$

Step 4: Substitute to Find Variance

Now we use the formula for variance:

[ \text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 ]

Substitute $\mathbb{E}[X] = \frac{\alpha}{\alpha + \beta}$ and $\mathbb{E}[X^2] = \frac{\alpha (\alpha + 1)}{(\alpha + \beta)(\alpha + \beta + 1)}$:

$\text{Var}(X) = \frac{\alpha (\alpha + 1)}{(\alpha + \beta)(\alpha + \beta + 1)} - \left( \frac{\alpha}{\alpha + \beta} \right)^2$

Step 5: Simplify the Expression

Rewrite $\left( \frac{\alpha}{\alpha + \beta} \right)^2$ as:

$\left( \frac{\alpha}{\alpha + \beta} \right)^2 = \frac{\alpha^2}{(\alpha + \beta)^2}$

Substitute back into the variance formula:

$\text{Var}(X) = \frac{\alpha (\alpha + 1)}{(\alpha + \beta)(\alpha + \beta + 1)} - \frac{\alpha^2}{(\alpha + \beta)^2}$

To combine these terms, get a common denominator of $(\alpha + \beta)^2 (\alpha + \beta + 1)$:

$\text{Var}(X) = \frac{\alpha (\alpha + 1)(\alpha + \beta) - \alpha^2 (\alpha + \beta + 1)}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$

Step 6: Expand and Simplify the Numerator

Expanding the terms in the numerator:

[ = \frac{\alpha (\alpha^2 + \alpha \beta + \alpha + \beta) - \alpha^2 (\alpha + \beta + 1)}