Wright-Fisher model Consider N haploid individuals, each carrying 1 copy of a specic genetic locus (a location of interest in the genome]. Each individual (= gene) can be of two types (= alleles), A or a, which correspond to two different pieces of genetic information at the same locus. Suppose that at each time unit each individual randomly and uniformly chooses another individual (possibly itself ) from the population and adopts its type (\"parallel updating\"). This is called resampling, and is a form of random reproduction. Suppose that all individuals update independently from each other and independently of how they updated at previous times. We are interested in the evolution of the following quantity: X\" = number of A's at time n. Example 1. Let N = 6 and denote an individual of type A by 0, while an individual of type a by a. We would like to compute the probability that there will be 3 individuals of type A at time in. given that there are 4 individuals of type A at time (n 1), Le. P(X = 3 I X _1 = 4). First, there are [3) possible choices of type A individuals in the n'\" time ame. Next, the probability of choosing an individual of type A for the update is g: g and of type 0., hence, P(X,, = 3 IX _1= 4) = (g) [as (as. . . U . . . . . . X. (a) [5 pts] Let Al: be the state with 3' individuals of type A [here i = O,1,2,3). Show that the process {X0,X1,X2,. . . ,} is a Markov chain1 and list the absorbing states. nl Problem 1 We consider the case N = 3. (b) [5 pts] Compute the transition probabilities. from/to A0 A1 A2 A3 Denition 2. A discrete-time stochastic process {Y0,Y1,Y2, .. .} is called a martingale if E(Y,,|Y -1, .. .,Y0) = Yn_1 for any time n 2 I]. In other words, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation