Homework3:


Dirichlet distributions.



1. Suppose X1, X2, ... Xk  are iid U(0,1) random variables.
Order these observations: 0 < X(1) < X(2) ... < X(k) < 1

Define the (K+1) random variable as the "gap times":

Y1 = X(1) - 0
Y2 = X(2) - X(1)
Y3 = X(3) - X(2)
.......
YK+1 = 1 - X(k)

Find the joint density of the Y vector. (Apparantly those k+1 gaps add up to one, so 
you should work on the k gaps joint distribution)



2. If for i=1,2,... k; we have independently random variables
X_i ~ Gamma( shape = alpha_i, scale =1)

Show that the joint density of 

Y_i = X_i / sum_j X_j

is Dirichlet.