[s:16]讲马科夫链（P.192）的例子4.10
Example 4.10
A pensioner receives 2 (thousand dollars) at the beginning of each month. The amount of money he needs to spend during a month is independent of the amount he has and is equal to i with probability Pi, i = 1, 2, 3, 4,P1+P2+P3+P4 = 1. If the pensioner has more than 3 at the end of a month, he gives the amount greater than 3 to his son. If, after receiving his payment at the beginning of a month, the pensioner has a capital of 5, what is the probability that his capital is ever 1 or less at any time within the following four months?
Solution: To find the desired probability, we consider a Markov chain with the state equal to the amount the pensioner has at the end of a month. Because we are interested in whether this amount ever falls as low as 1, we will let 1 mean that the pensioner’s end-of-month fortune has ever been less than or equal to 1. Because the pensioner will give any end-of-month amount greater than 3 to his son, we need only consider the Markov chain with states 1, 2, 3 and transition probability matrix Q = [Qi,j ] given by
|| 1 0 0 ||
|| P3+P4 P2 P1 ||
|| P4 P3 P1 + P2 ||
To understand the preceding, consider Q2,1, the probability that a month that
ends with the pensioner having the amount 2 will be followed by a month that
ends with the pensioner having less than or equal to 1. Because the pensioner
will begin the new month with the amount 2+2 = 4, his ending capital will be
less than or equal to 1 if his expenses are either 3 or 4. Thus, Q2,1 = P3 + P4.
The other transition probabilities are similarly explained.
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以上是书中的描述，我的疑问是状态应该写成：
|| P2+P3+P4 P1 0 ||
|| P3+P4 P2 P1 ||
|| P4 P3 P1 + P2 ||
从状态1变成状态1的概率应该是P2+P3+P4，而不是P1+P2+P3+P4啊？？？
