H
hotsauce
Member
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?
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D
didster
Member
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.
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.
N
neha.neu
Member
Hi
I do not understand how the 45 term comes? also, why is it x(10-x)/45??
I do not understand how the 45 term comes? also, why is it x(10-x)/45??
M
Mark Mitchell
Member
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.
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.
R
randomWalker
Member
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)
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)