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Distributionshard
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Given X∼Gamma(α,β)X \sim \text{Gamma}(\alpha, \beta)X∼Gamma(α,β) and Y∼Gamma(γ,β)Y \sim \text{Gamma}(\gamma, \beta)Y∼Gamma(γ,β) are independent, the distribution of Z=XX+YZ = \frac{X}{X+Y}Z=X+YX​ is Beta(α,γ)Beta(\alpha, \gamma)Beta(α,γ). What is the variance of ZZZ?