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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)}

Prove that the variance of a beta-distributed random variable with parameters α and β is \( \sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} \).

Learn how to prove the variance of a beta-distributed random variable with parameters α and β. This solution covers key concepts in probability and statistics, including the mean and variance formulas for the beta distribution. Suitable for college-level students.

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