student number = 1716743

2. a. Consider a random walk S, = >_, Xi, with X, being i.i.d random vari- ables distributed according to the following rules: P[X = 1) = P[X = -1) = 1/2. The walk starts at So = N, where N is a positive integer, and the boundary at 0 is reflecting: for any time n such that S, = 0, the next step is +1 with probability 1. Using the method of first-step decomposition, find the expected number of steps until the walker reaches the position cN, where c is the fifth digit of your student number or the 2/5 number 2, whichever is larger. For full credit, the difference equation you use has to be explicitly derived. The rules for transforming the resulting conditional expectations can be applied without proof. [15 marks] b. Consider a branching process Wn+1 = EM Zi, where Z; (the number of offspring of the i-th individual) are i.i.d. random variables distributed according to the following rules: P[Z = 0] = 2/3, and for any n > 1, P[Z = n] = Co", where C and a are positive constants. Determine the value of C, and the extinction probability for this process if a = 2/3. [5 marks]