PLEASE POST THE SOLUTION IN A HANDWRITTEN FORMAT. THANK YOU.

PLEASE POST THE SOLUTION IN A HANDWRITTEN FORMAT. THANK YOU

4. This problem is about a continuous time branching process. This is analogous to the Galton-Watson-Bienaym process I used at the start of the course. Let X(t) denote the size of a population at time t so that X(t){0,1,}. Fix a time t and think about 1 individual in the current population. There is a chance h+o(h) that that individual will give rise to a random number of offspring; if J is the random number of offspring then P(J=j)=pj for each j>0. There is also a chance h+o(h) that the invidual will die. All individuals act independently. (a) The process X is a birth and death process only in one special case. What must the pj s be for X to be a birth and death process? (b) Give formulas for the entries in R, the infinitesimal generator. (c) Give lines 0,1 , and 2 of the transition matrix of the skeleton chain.

4. This problem is about a continuous time branching process. This is analogous to the Galton-Watson-Bienaym process I used at the start of the course. Let X(t) denote the size of a population at time t so that X(t){0,1,}. Fix a time t and think about 1 individual in the current population. There is a chance h+o(h) that that individual will give rise to a random number of offspring; if J is the random number of offspring then P(J=j)=pj for each j>0. There is also a chance h+o(h) that the invidual will die. All individuals act independently. (a) The process X is a birth and death process only in one special case. What must the pj s be for X to be a birth and death process? (b) Give formulas for the entries in R, the infinitesimal generator. (c) Give lines 0,1 , and 2 of the transition matrix of the skeleton chain