Note: Partial credit will be given if youi Total Points: 100 1. (25 points) Bob

Question

Note: Partial credit will be given if youi Total Points: 100 1. (25 points) Bob credit will be given if you indicate the correet approa en if you indicate the correct approach. process class and each week, his perfioemance eculd bs is above avernge in cee week, he will is taking a stochastic process class and each week, his ps above average, a be average or behind in ge, average, or behind for each week. If he is above aver the next week with probability 0.3 or oa will be above average or behind in the nest weck with probability respectively. If he is in one week, he 0.3 or one week, he will be above average or average in the next week with probability o.I and 0.6 respectively. To be "above average" state 2 to be "average" and state s to is given as below model the problem, we set state 1 to "average" and state 3 to be "behind". The transition matris [S e two-step and three-step transition matrices are given as below P2 The 360 0.300 0.260 0.390 0.430 0.450 0.250 0.270 0.290 0.264 0.322 0.306 0.294 0.414 0.424 0.432 P3 0.270 0.274 If Bob is equally likely to be above average, average, and behind at week 0, what's the probability that he will be above average at week 2? (6 points) 1)

Explanation / Answer

1.) Probability of being above average in 2 weeks to will depend on the second order transition matrix, as theier is equal probitity for being at any stage now so, the answer will be 1/3*.360+1/3*.300+1/3*260, that is summation of probability of being at a particular state in time 0* probability of transfer from that stage to above average in 2 weeks of time.=0.30667

2.) Expected no. weeks = summation of number of weeks he will be above average *probability of transfer from below to above average =1*0.1+2*0.266+3.*0.294=1.502

3.)=Probability thst he will be above average in week 1* probability that he will go from above average in week one to behind average in week 2 =.360*.264=0.09504

4.) Let a,b, and c be the respective steady state probabilities for anbove average, average, and below average,then the equations are:

a=0.360a+.300b+.260c

b=0.390a+0.430b+0.450c

c=0.250a+0.270b+0.290c

and, a+b+c=1

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