Consider the weatherforecast model where S 0 1 where Xn is

Consider the weather-forecast model where S = (0, 1), where Xn is 0 or 1 according as it rains or does not rain in the n^th day. The transition matrix is given by Let X0 = 1 and define T1 = min{n geq 1: Xn = 1}. (a) Find the distribution T1. (b) Find expectation of T1.

Solution

T1 = min of Xn =1

X1 or x2 or x3,,,,,,,, can be equal to 1

Let us assume minimum is xn

Distribution of T1 =

Distribution of Xn =1

= 1-beta,

E(T1) = 1-E(Beta)