Word Count : 935

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In the study of probability distributions, the gamma distribution occupies an important position due to its flexibility and wide applicability in modeling waiting times, lifetimes, and sums of exponential random variables. When two independent gamma-distributed random variables are considered together, interesting derived distributions arise from their combinations. One such important result concerns the ratio of one gamma variable to the sum of two independent gamma variables, which leads to the beta distribution. Let two independent random variables (X) and (Y) follow gamma distributions with parameters (m) and (n) respectively, and a common scale parameter (\theta). In standard notation, this can be expressed as (X \sim \text{Gamma}(m, \theta)) and (Y \sim \text{Gamma}(n, \theta)). The probability density functions of (X) and (Y) are given by [ f_X(x) = \frac{1}{\Gamma(m)\theta^m} x^{m-1} e^{-x/\theta}, \quad x > 0 ] and [ f_Y(y) = \frac{1}{\Gamma(n)\theta^n} y^{n-1} e^{-y/\theta}, \quad y > 0, ] where (\Gamma(\cdot)) denotes the gamma function. Since (X) and _____ ______ ___ ____ _________ ________ ____ _______ ________ _______.
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