How is the generator matrix derived? Also, for (iii), how is the parameter calculated? It seems to be doing u * Pr(increase in cats with fleas). But what exactly does u represent in this problem?
Entries in generator matrix are derived as follows. x->x+1 is rate of moving from x to x+1, ie rate of one more cat getting infected. This is rate of two cats meeting, mu times probability that one of the cats is infected and the other is not, x(10-x)/45 (otherwise nothing changes) times the probability that the fleas jump and infection occurs, 0.5. IE, mu/45*x*(10-x)*.5 or mu/90 {(x)(10-x} The entries inside the matrix are the x(10-x). The x,x entries are simply the negative of the x,x+1 as the row must sum to zero (see below) All the others are zero as you can only go to x+1 if you are in x, ie one more gets infected if "good meeting" occurs, stay the same if not. u (mu?) is the instantaneous rate of cats meeting, ie parameter in Poisson process, waiting time are exponentially distributed with parameter u, etc.
In part (i) of the question we explain why the number of possible pairings which could result in a new flea infection is x(10-x) if x is the number of cats with fleas. To calculate a probability, we divide this by the number of possible pairings of 10 cats. This is 10_C_2 = 10!/(2!*8!) = 45, using combinatorics.
I know this is an old thread but I thought I'd post under same question. Just wanting to know if my intuition is correct on this one.. I found it really challenging and different to anything seen in the notes. If Mu_ij is the ith, jth element of generator matrix. And represents the rate of infection from state i to j. Oi = Probability that infection can occur, with i infected cats = (#Pairs resulting in new infection with i infected cats) / (Total # of possible pairings) Mu_ij = (Probability of Infection = 0.5) x Oi x ( Rate of cats meeting = mu)
