Suppose whether the stock market index S&P 500 goes up or down in value tomorrow depends on its performance today and yesterday. In particular, we have learned if S&P 500 went up today and yesterday, it will go up tomorrow with probability 0.7 if S&P 500 went down today and yesterday, it will go up tomorrow with probability 0.1 if S&P 500 went down yesterday but went up today, it will go up tomorrow with probability 0.45 if S&P 500 went up yesterday but went down today, it will go up tomorrow with probability 0.3 Define the state of the S&P 500 index such that its behaviour can be described by a Markov chain. give the transition probability matrix for the chain. Suppose whether the stock market index S&P 500 goes up or down in value tomorrow depends on its performance today and yesterday. In particular, we have learned if S&P 500 went up today and yesterday, it will go up tomorrow with probability 0.7 if S&P 500 went down today and yesterday, it will go up tomorrow with probability 0.1 if S&P 500 went down yesterday but went up today, it will go up tomorrow with probability 0.45 if S&P 500 went up yesterday but went down today, it will go up tomorrow with probability 0.3 Define the state of the S&P 500 index such that its behaviour can be described by a Markov chain. give the transition probability matrix for the chain