Let{Xn,n0}be a random walk which describes the position of an object at timenthat starts at 0 and at each step goes either uphunits with probabilitypor downhunits with probability 1p.

1. a)Show that ifXnis a standard random walk, i.e. ifp=1, thenXnis a martingale.
2. b)Show that ifp=1, thenXnis not a martingale.
3. c)LetYn=XnXn1. Calculateg() =E(eYn).
4. d)Show thatMn=eXnnlogg()is a martingale.
5. e)Showthatifh=1,then(1p/p)^Xn,n1,isamartingale.