5. SIGMAALGEI 3m (40 POINTS, EQUALLY DISTRII 3UTED) Let the price of one share of a stock be represented by a stochastic process X = {Xi : t E {0, 1, 2, 3. 4}}. Consider the following possible realizations for the stochastic process X0011) X1021) X2(w) X300) X400) (01 11 6612 11 013 11 {04 11 M5 11 (06 11 (1)7 11 M8 11 Mg 11 M10 11 0111 11 0112 11 (1)13 11 U114 11. (1)15 11 U116 11. ) ) 3) 1 13 13 13 13 13 13 13 13 13 13 11 11 11 11 11 11 15 15 15 15 15 15 12 12 12 12 12 12 12 11 11 11 16 16 15 15 14 14 13 13 13 13 13 13 12 14 14 12 16 14 16 14 16 14 14 18 12 11 14 12 12 20 12 12 FIGURE 1. Realizations of a stochastic process \"What is the conditional distribution of X4 if you know X2? Calculate the conditional expectation of X4 if you know X2. IP' (60) 0,06 0,065 0,06 0,065 0,06 0,065 0,0625 0,08 0,04 0,0675 0,0625 0,0625 0,0625 0,0625 0,0625 0,0625 1 \"What cralgebras (F0, .71, F2) are generated by random variables X0, X1, and X2? 4 This exercise explains why ltration .70 C .71 C .72 is called information. Assume t : 1. List some events from 3:2 occurence of which you currently cannot determine