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We need to find the distribution of \( U = \frac{X}{X + Y} \) when \( X \sim \text{Gamma}(\alpha, \lambda) \) and \( Y \sim \text{Gamma}(\beta, \lambda) \), given that they are independent. ### Step 1: Understanding the Joint Distribution of \(X\) and \(Y\) The probability density function (PDF) of a Gamma-distributed random variable \( X \sim \text{Gamma}(\alpha, \lambda) \) is: \[ f_X(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}, \quad x > 0 \] Similarly, for \( Y \sim \text{Gamma}(\beta, \lambda) \): \[ f_Y(y) = \frac{\lambda^\beta}{\Gamma(\beta)} y^{\beta - 1} e^{-\lambda y}, \quad y > 0 \] Since \( X \) and \( Y \) are independent, _________ ________ ______ ________ ____ ___ __________ ___ _______.
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