12. Suppose (X1,. .. , Xn) is a random vector as in the above exercise having a Dirichlet distribution with parameter vector (@1, . .. , On+1). Then, let us define new random variables Y1, . . . , Yn as follows: X1 Y -1-X1-.. . - Xn Yn = ; Xn 1-X1-. . . - Xn (a) Derive the joint density function of (Yi, . . . , Yn). This is what is called the Dirichlet density of the second kind; (b) Find the marginal density function of Yi for i = 1, . . . , n; (c) Derive explicit expressions for the mean vector and variance-covariance matrix of (Y1, . . . ,Yn)