**Proof of the Variance of a Beta-Distributed Random Variable**The variance of a beta-distributed random variable with parameters \(\alpha\) and \(\beta\) is given by:\[\sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\]To prove this formula, one must derive the expression starting from the definition and properties of the beta distribution. Here, \(\alpha\) and \(\beta\) are shape parameters of the distribution.

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A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Prove that the variance of a beta-distributed random variable with parameters a and 3 is αβ (a + B)² (a + B + 1)* =