Note that patients may receive this treatment for multiple days and therefore be in- cluded in this count data on multiple days. (This time series does not give new admissions to ventilation or the number leaving ventilation each day.) A time series model to estimate the rates of daily admissions and daily discharge from ventilation is constructed in this assignment. The model is called the \"integer autoregrassion" and is taken from the eld of queuing theory (the study of lengths of queues). Let Y; denote the number of patients receiving ventilation on day t (i.e. Vent). This is modelled as the sum of two independent components 0 the number of patients who were receiving ventilation on day t 1 not yet discharged on day t, denoted X;, o the number of new arrivals to ventilation treatment, denoted Ut, so that 1/; can be represented K = X; + Ug. We will model Ut as being i.i.d. with Poisson distribution with parameter A, independent of Xt. The model for X; is constructed as follows. Each of the YLI patients receiving ventila- tion at time t 1 are modelled as having probability p of remaining on ventilation at time t. Each of these patient outcomes (remain / leave ventilation) are considered independent of all other patients. 1. Conditional on 11-1, what is the distribution of Xt? (Hint: look back to distributions from QMl!) 2. Use question 1, the Poisson assumption for U; and the convolution formula1 to give an expression for the probability mass function of Y; conditional on 14-1. 3. Why does this model merit the term \"autoregression\"? (Hint: derive E(Yt|.77t_1)) 11f 21 and Z2 are independent integer random variables and Z = Z1 + Z2 then P(Z = z) = 215121 = J')P(Zz = z 3') i=0