The basic distribution theory in this question underlies the discount volatility model of Section 4.3.7 and the results to be shown below in Problem 11. Two positive scalar random quantities 0 and 1 have a joint distribution under which:

0 G(ab) for some scalars a > 0b > 0; and p( 1 0) is implicitly de ned by 1 = 0 where with independent of 0 and where Be( a(1

)a)

(01) is a known, constant discount factor.

(a) What is E( 1 0)?

(b) What are E( 0) and E( 1)?

(c) Starting with the joint density p( 0)p( ) (a product form since 0 and are independent), make the bivariate transformation to ( 0 1) and show that p( 0 1)=c e b 0 a 1

( 0 1)(1 )a 1 on 0< 1< 0 1

being zero otherwise. Here c is a normalizing constant that does not depend on the conditioning value of 0

(d) Derive the p.d.f. p( 1) (up to a proportionality constant). Deduce that the marginal distribution of 1 is 1 G( a b)

(e) Show that the reverse conditional p( 0 1) is implicitly de ned by 0 = 1+ where G((1 )ab)

with independent of 1