The random variables X,, X, are said to be exchangeable if X,, ., X, has the same joint distribution as X,., X, whenever i, i, , i, is a permutation of 1, 2, , n That is, they are exchangeable if the joint distribution function P{X, x, X x2, ..., X, x} is a symmetric function of (x, x2, .,x,). Let X1, X2, denote the interar- rival times of a renewal process

(a) Argue that conditional on N(t) = n, X, ., X, are exchangeable Would X, X, X., be exchangeable (conditional on N(t) = n)?

(b) Use

(a) to prove that for n > 0 X + E + XN (1) N(t) N(t) = n =E[X|N(t)=n].

(c) Prove that X + + X N(O) E N(t) | N() > 0] = N(t)>0 E[X| X