Urgent query on Chapter 5

[Goku]

Hi everyone,

Please have a look at the example that follows Section 4.2 (The Integrating Factor Method) in Chapter 5 of the notes. The example says:
'It can be shown that: d/dt tPx ^ (SM) = ...'
Upon trying to show (on my own) that this indeed is the solution, I found that the term containing the transition probability from single to divorced/separated is missing. Clearly, this is a possible transition and should be included as one of the terms, after using the Kolmogorov Forward equations. However, it is NOT there in the solution provided by the notes.

Please students, have a look at this and let me know what your'll think. Acted Tutors, please look at this as well, and please provide a comprehensive explanation why this is so IF the notes are correct (which I hope are not, as this will really confuse things for me then in this section...it will go against my logic gained in this section). Also, if the solution given in the notes does turn out to be an error, you can perhaps add it into the FAQ thread.

Sincere thanks,

Happy Studying!

 

[Erik]

Hi Goku,

I didn't look at the example and passed CT4 long ago.
But, I think the reasoning is that you can't go directly from single to divorced/seperated. You must firts go to married! This is why there is no transition between single to divorced.

Remember with continuous time processes you only look at the transitions that can occur instantaneously, not those that may occur over a long period of time. So even though it is possible to go from single to divorced over a longer period of time, you can't do so instantaneously!

I hope this helps and remember to watch out for these type of things when they ask you to draw the transition graph.

Regards,
Erik.

 

[Goku]

Hi Erik,

I thought about it, but then using the same argument you proposed (instantaneous transitions), the probability that a state makes a transition to ANY state in an instant is always zero.

I'm still confused, but thank you for the reply.

Goku

 

[Erik]

Hi Goku, I know that it is difficult to understand.

You need to argue like this. When a person gets married, it is a transition that CAN happen in an instant. So the the probability of transitioning over an instant is zero (or close to), that's why we use transition rates instead of transition probabilities as in Markov chains.

This means that in an instant (a very small period of time, say one second) you can go from being single to married or from married to divorced. But you cannot go from single to divorced in one second, because you must first get married!

Maybe one of the ACTED tutors can help you further with this explanation, but I am 100% convinced that you can't go directly from Single to Divorced. It doesn't make sense.

Regards,
Erik.

 

[Goku]

Hi Erik!

I understand fully your argument and would love to be perfectly comfortable with it - it's just that, by derivation, those Kolmogorov equations allow for any state to be entered into in a transition from i to j, except the jth state, when finding d/dt tPx ^ (ij). Also there isn't a 'h' time period (a very small period) over which these transitions can take place, which, if there was, would then possibly validate your argument.

You are 100% correct! There can NEVER be a transition rate from single to divorced. But it is possible to have a non-zero probability from single to divorced over a time t, and from there an instantaneous transition rate from divorced to married i.o.w tPx ^ (SD) . u(x+t) ^ (DM) is perfectly possible.

I hope you understand what I'm saying. Thank you, though, for you interest in my query! Please, ActEd tutors, engage thyselves in this query of mine.

Thank you,
Goku

 

[Goku]

ActED tutors,

I humbly ask once again to please assist in this query.

Thank you,
Goku

 

[Julie Lewis]

Goku

You are absolutely right - there is a term missing on the rhs of that differential equation.

It is tpx(SV) * gamma.

This represents the probability of going from single at age x to divorced at age x+t and then doing an instantaneous transition from divorced to married at age x+t.

Unfortunately, this means that the differential equation cannot be solved using the integrating factor method unless we assume that gamma = 0.

Apologies for the error. I will get the correction added to the CT4 corrections document. The thing that worries me is that this error must have been in the course for at least a couple of years and none of you have pointed it out to us until now!!!

 

[Goku]

Thank you

Thank's Julie! I'm glad that this query has finally been put to bed - I'm sure many more students would have noted this.

Goku