Question: .Here, we derive the Beta distribution. Consider X1 ~ Gamma(a, 1) and X2 ~ Gamma(8, 1) independent. We will show that the distribution of Y

.Here, we derive the Beta distribution. Consider X1 ~ Gamma(a, 1) and X2 ~ Gamma(8, 1) independent. We will show that the distribution of Y = X1/(X1 + X2) follows the Beta(a, B) distribution. (a) Consider the change of variables from (X1, X2) to (Y, Z) where Z = X1 + X2. Find the inverse functions and the Jacobian. (b) Using the bivariate change of variables results, find the joint distribution of Y and Z. (c) Find the marginal distribution of Y and show that this matches with the Beta(a, ) distribu- tion. (d) Show that Y and Z are independent and find the distribution of Z

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