b. Let X- (n=1,2. ...) denote the rainfall level at a particular site on day n. In a simple model, X- takes one of the three values. If there is no rain on day n, then X- = 1 . If there is light (that is, total rainfall is below some fixed amount (c)) then, X_ = 2 , while X- =3 if the rainfall is heavy (when total rainfall is c or more). Some data have been collected for the period between December 1 to February 28 for five years, from which the following table has been constructed. Day n+1 1 2 3 224 39 26 Day n 31 43 23 32 17 10 The first row of the table gives the number of dry days, which were followed respectively by a dry day (column 1), a light rain day (column 2), and a heavy rain day (column 3), similarly to the other rows. 1) On the assumption that X- is a Markov chain, use the above data to estimate the one-step transition probability matrix for the chain. 11 If January 3 is a dry day, find the probability that: a) there will be a heavy rain on January 4 () there will be light rain on January 5