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 (Y) are independent, their joint density function is simply the product of the individual densities: [ f_{X,Y}(x,y) = \frac{1}{\Gamma(m)\Gamma(n)\theta^{m+n}} x^{m-1} y^{n-1} e^{-(x+y)/\theta}, \quad x>0, y>0. ] Now consider a transformation of variables that allows us to study the ratio of interest. Let us define two new random variables: [ U = X + Y, \quad V = \frac{X}{X + Y}. ] Here, (U) represents the total sum of the two gamma variables, and (V) represents the proportion contributed ________ ____ _____ _____ ________ ______ ___ ____ _________ _______ _______.
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