These are the problems to solve.

1. Let X and Y be independent and uniform on {1, . .., 5}. Find a) the conditional distribution b) the conditional expectation of X given X - Y = 1.5. Let (Xn) be a simple random walk on Z starting in 0. Which of the fol- lowing are martingales, and why? b) (3X)2 - 9n c) Ek=3(Xk-3 - 3Xk-1Xk-2)(Xk - Xk-1)6. Let En,i, n = 1,2,...,i = 1,2. .... be i.i.d No-valued random variables with E[1,1] = 2. Define inductively Xo :=1, Xn+1 := EnEn,i. a) Find E[Xn+1|(Xo, . .., Xn)]. b) Give an argument why (Ca ) converges a.s. to an integrable random va- riable X as n - 00.4. In the compound Poisson process Zt = >are Hi, t 2 0, let (Ti, T2, ... ) be a standard Poisson process and let the H, have a normal distribution with mean 2 and variance 3. Compute the expectation and the variance of Zt- Im Compound Poisson Prozess Zt = Ciret Hi, t 2 0, sei (Ti, T2, ...) ein Standard-Poissonprozess, und jedes H; habe eine Normalverteilung mit Er- wartungswert 2 und Varianz 3. Berechnen Sie Erwartungswert und Varianz von ZA.2. Let (71, 72, ..., ) be a standard Poisson process on Ry, and N its number of points up to (and including) t. a) For k c N, express the event (7k+1 > {} \\{Th > {} in terms of Ne and k. b) For k E N, let Gx be Gamma(k)-distributed. Find P(Gk+1 > () - P(GR > (). (Hint: You can answer this without any calculation.)3. Let X = (X,) be the nearest neighbour random walk on Z with upwards probability p = 3/4. a) For which number v is Xn - on a martingale? b) Compute the expected value of the time T when X first hits state 10, given that X starts in state 0. Hint: What does the stopping theorem tell for this 7? You may use without proof that (XTAn) is uniformly integrable. Sei X = (Xn) die Nachste-Nachbar-Irrfahrt auf Z mit W'keit p = 3/4 fur einen Schritt nach oben. a) Fur welche Zahl v ist X, - on ein Martingal? b) Berechnen Sie den Erwartungswert der Zeit T, zu der X bei Start im Zustand 0 erstmals den Zustand 10 trifft. Hinweis: Was sagt der Stoppsatz fur dieses 7? Sie durfen ohne Beweis ver- wenden, dass (XTAn) uniform integrierbar ist