Exercises marked with a ” ” are programming exercises. These can be solved in a programming lan- guage of your choice. Please make sure to hand in your code along with your answers to these exercises.
Problem 1: Random walk (tutorial exercise)
Consider a symmetric, time-discrete random walk in one dimension, {X(t) | X(t) ∈ Z,t ∈ N0}. Let a/2 denote the probability of jumping forward or backward, respectively. Hence we have transition probab- ilities Pi,i+1 = Pi,i−1 = a/2, Pi,i = 1 − a and otherwise Pi, j = 0.
Hint: Express X(t) as the sum of increments ∆(t) ∈ {−1,0,+1}, i.e. X(t) = X(0)+∆(1)+···+∆(t) and use that the ∆(t) are iid (identically independently distributed).
- (a) Argue why E[∆(t)] = 0.
- (b) Argue why E[X(t)] = x0, where x0 = X(0).
- (c) Calculate the variance Var[∆(t)]. Hint: Var[X] = ∑i=1 pi ·(xi −xˆ)2.
- (d) Show that the variance of X(t) equals Var[X(t)] = at.
Problem 2: Neutral Moran process
Consider the neutral Moran process {X(t) | t = 0,1,2,…} with two alleles A and B, where X(t) is the number of A alleles in generation t.
(a) Show that the process has a stationary mean: (1 point) E[X(t)|X(0)=i]=i.
Hint: First calculate E[X(t) | X(t −1)] and use the law of total expectation, EY [Y] = EZ[EY [Y | Z]] withY =X(t)andZ=X(t−1).
(b) Show that the variance of X(t) is given by:
Var[X(t) | X(0) = i] = V1 2/N2 .
Consider the following steps:
(i) Show that V1 := Var[X(1) | X(0) = i] = 2i/N(1−i/N).
(2 points)
(1)
1−(1−2/N2)t
(ii) Then use that ∀t > 0 Var[X(t) | X(t −1) = i] = Var[X(1) | X(0) = i] (why?) and the law of totalvariance,Var[Y]=EZ[VarY[Y |Z]]+VarZ[EY[Y |Z]],toderive
Var[X (t ) | X (0) = i] = V1 + (1 − 2/N 2 ) Var[X (t − 1) | X (0) = i] (2) 1
