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Jy88
04-25-2007, 01:26 AM
This is from Actex practice problem 34.

Given that the probability of switching to regime 2, conditional on being in regime 1 is 0.1. The probability of switching to reigme 1, conditional on being in regime 2 is 0.2.

Hence, we are asked to calculate the unconditional probability of spending k months in regime 1, where k=0,1,2.

In the solution provided:

Based on the recursive relationship:

P[Rn(t)=r|p(t-1)] = p(t-1,1)* P[Rn(t+1)=r-1|p(t)=1]+
p(t-1,2)* P[Rn(t+1)=r|p(t)=1]

Now, why would P[R2(0)=0|p(0)=1]=0.1*0.8? and the same goes with the following:
P[R2(0)=0|p(0)=2] = 0.8*0.8
P[R2(0)=1|p(0)=1] = 0.1*0.2+0.9*0.1
P[R2(0)=1|p(0)=2] = 0.8*0.2+0.2*0.1
P[R2(0)=2|p(0)=1] = 0.9*0.9
P[R2(0)=2|p(0)=2] = 0.2*0.9

My understanding is P[R2(0)=0|p(0)=1] is the probability that given now is at regime 1, it will spend 0 periods in regime 1. how do we use the recursive formula to get the above probabilities?

Thanks a lot for that.

bagheera
04-25-2007, 05:30 AM
This is from Actex practice problem 34.

Given that the probability of switching to regime 2, conditional on being in regime 1 is 0.1. The probability of switching to reigme 1, conditional on being in regime 2 is 0.2.

Hence, we are asked to calculate the unconditional probability of spending k months in regime 1, where k=0,1,2.

In the solution provided:

Based on the recursive relationship:

P[Rn(t)=r|p(t-1)] = p(t-1,1)* P[Rn(t+1)=r-1|p(t)=1]+
p(t-1,2)* P[Rn(t+1)=r|p(t)=1]

Now, why would P[R2(0)=0|p(0)=1]=0.1*0.8? and the same goes with the following:
P[R2(0)=0|p(0)=2] = 0.8*0.8
P[R2(0)=1|p(0)=1] = 0.1*0.2+0.9*0.1
P[R2(0)=1|p(0)=2] = 0.8*0.2+0.2*0.1
P[R2(0)=2|p(0)=1] = 0.9*0.9
P[R2(0)=2|p(0)=2] = 0.2*0.9

My understanding is P[R2(0)=0|p(0)=1] is the probability that given now is at regime 1, it will spend 0 periods in regime 1. how do we use the recursive formula to get the above probabilities?

Thanks a lot for that.

|p(0)= should be replaced by |p(-1)= everywhere because you are "starting" from t=0.
You can't use recursion for P[R2(0)=0|p(-1)=1] because r=0 here and the recursion involves r-1. Here you just need to find the probability that between t=0 and t=2 (2 not inclusive, it's just 0 and 1) you are never in regime 1 given that you are in 1 at t=-1. So it is just the probability that you move to 2 (probability=0.1) *and* stay there
(probability=0.8). So the probability of the "and" is 0.1*0.8.
Whenever r=0 you cannot use recursion, and you don't need to use recursion either. Just use the reasoning above.

Let's now see P[R2(0)=1|p(0)=2] for example. It should actually be
P[R2(0)=1|p(-1)=2].
P[R2(0)=1|p(-1)=2]=p_21 * P[R2(1)=0|p(0)=1]+p_22*P[R2(1)=1|p(0)=2]

= 0.2* p_12 + 0.8* p_21
=0.2*0.1+0.8*0.2 (possible paths are 2->1->2 and 2->2->1)

The other cases can be similarly handled. But I would suggest you not use recursion at all for a 2 or 3 step problem. Just determine the possible ways/paths to achieve the result and write down the corresponding products of probabilities.