statistics

3) Consider a system in which individuals at any time are classified as being in one of 1' possible states, and assume that an individual changes states in according with a Markov chain having transition probabilities Pij- i,j : 1, . . . , 1-. That is, if an individual is in state 71 during a time period then, indpenedently of its previous states, it will be in state 3' during the next time period with probability Hj- The individuals are assumed to move through the system independently of each other. Suppose that the numbers of people initially in states 1, . . . , r are independent Poisson random variables with respsectve means A1,...,A,.. For fixed 3', let MG), j = 1, . . . ,r denote the number of those individuals, initially in state i, that are in state j at time 72.. Now each of the (Poisson distributed) number of people initially in state i will, independently of each other, be in state j at time n with probability Pg, where Pf;- is the n-state transition probalbility for the Markov chain having transition probabilities Rj. Let X,- be the number of individuals at the initial status for each state j:1,...,r. (a) Then, we can show that MW : I1 + I2 + - - - + IX\" where 1;, : 1, if the k-th individual initailly in state 2' is in state 9' at time n; 0, otherwise. What is the distribution of Ik? What is the conditional distibution of Nj) given Xi? (2pt) (b) Derive the maginal distribution of Njl). [Hintz X,- N Poisson(,\\,;)] (3pt)