Let r_ij(n)=P(X_n = j | X_0=i) be the n-step transition probability of a given homogeneous discrete-time Markov chain with m states. We have shown that r_ij(n) satisfies the following recursion for n>=2:
r_ij(n) = sum_(k=1)^(m) r_ik(n-1)*p_kj
For each of the following, decide whether it is also a valid recursion formula for r_ij(n).
1. r_{ij}(n)=\sum _{k=1}^ m p_{ik}r_{kj}(n-1) for n>=2
2. r_{ij}(n)=\sum _{k=1}^ m r_{ik}(n-2)r_{kj}(2) for n>=3
3. r_{ij}(n)=\sum _{k=1}^ m \sum _{\ell =1}^ m r_{ik}(n-2)p_{k\ell }p_{\ell j} for n>=3
yes,
yes
yes
Hmm, let me put on my bot thinking cap for a moment. For a clown like me, these formulas look a bit serious, but I'll do my best to give you an answer with a touch of humor!
1. r_{ij}(n) = \sum_{k=1}^m p_{ik} r_{kj}(n-1) for n >= 2
Ah, this one is quite a straightforward one! It's like saying that the probability of going from state i to state j in n steps is equal to the sum of the probabilities of going from state i to all other states k, and then from state k to state j in n-1 steps. It's like taking a little detour through all the other states before reaching j. So yes, this is a valid recursion formula for r_ij(n)!
2. r_{ij}(n) = \sum_{k=1}^m r_{ik}(n-2) r_{kj}(2) for n >= 3
Oh, now this one seems a bit more complicated! It's like saying that the probability of going from state i to state j in n steps is equal to the sum of the probabilities of going from state i to all other states k, but with a twist! We have to take into account the probabilities of going from state i to state k in n-2 steps, and then from state k to state j in exactly 2 steps. It's like doing a little dance between the states before reaching j. Well, as much as I love a good dance, I'm not sure if this formula holds true for r_ij(n), so I'll pass on this one.
3. r_{ij}(n) = \sum_{k=1}^m \sum_{\ell=1}^m r_{ik}(n-2) p_{k\ell} p_{\ell j} for n >= 3
Oh my, this formula looks like a real party! It's like saying that the probability of going from state i to state j in n steps is equal to the double sum of probabilities. First, we need to go from state i to all other states k in n-2 steps with a probability of r_{ik}(n-2). Then, for each k, we need to go from state k to all other states \ell in one step, weighted by p_{k\ell}, and finally from state \ell to state j in one step, weighted by p_{\ell j}. So, in a way, it's like throwing a double sum party to get from i to j. Well, I have to say, this formula seems quite valid for r_ij(n)!
I hope my answers entertained you while shedding some light on these recursion formulas. Just remember, in the world of Markov chains, even the formulas like to have a little fun!
To determine whether each of the given formulas is a valid recursion formula for r_ij(n), we will compare them to the original recursion formula r_ij(n) = \sum _{k=1}^ m r_ik(n-1)p_kj.
1. r_{ij}(n)=\sum _{k=1}^ m p_{ik}r_{kj}(n-1) for n>=2:
This formula is a valid recursion formula for r_ij(n). It follows the same structure as the original recursion formula, but the roles of i and k are switched. Instead of summing over the states j, we are summing over the states k. The recursion formula states that the n-step transition probability from i to j is determined by summing the product of the one-step transition probability from i to k and the (n-1)-step transition probability from k to j.
2. r_{ij}(n)=\sum _{k=1}^ m r_{ik}(n-2)r_{kj}(2) for n>=3:
This formula is not a valid recursion formula for r_ij(n). It does not follow the same structure as the original recursion formula. While it involves the (n-2)-step transition probability r_ik(n-2) and the 2-step transition probability r_kj(2), it does not account for the one-step transition probability p_kj. The formula suggests that the n-step transition probability from i to j is determined by summing the product of the (n-2)-step transition probability from i to k and the 2-step transition probability from k to j, without considering the one-step transition probability.
3. r_{ij}(n)=\sum _{k=1}^ m \sum _{\ell =1}^ m r_{ik}(n-2)p_{k\ell }p_{\ell j} for n>=3:
This formula is a valid recursion formula for r_ij(n). It follows a similar structure as the original recursion formula. The double summation involves summing over the states k and l, similar to the original formula's summation over the states k. The term r_ik(n-2) represents the (n-2)-step transition probability from i to k, and the product of p_kl and p_lj accounts for the one-step transition probability from k to l and the one-step transition probability from l to j, respectively.
In summary, option 1 and option 3 are valid recursion formulas for r_ij(n), while option 2 is not a valid recursion formula.
To determine whether each of the given formulas is a valid recursion formula for r_ij(n), we can compare them to the original recursion formula for r_ij(n), which is:
r_ij(n) = sum_(k=1)^(m) r_ik(n-1)*p_kj
Let's analyze each formula individually:
1. r_{ij}(n) = sum_(k=1)^(m) p_ik * r_kj(n-1) for n >= 2
To compare this formula to the original one, we can see that the terms are just swapped compared to the original recursion formula. Instead of having r_ik(n-1) as the coefficient and p_kj as the probability for each term, we now have p_ik as the coefficient and r_kj(n-1) as the probability. However, the sum is still over all k from 1 to m. Therefore, this formula is a valid recursion formula for r_ij(n).
2. r_{ij}(n) = sum_(k=1)^(m) r_ik(n-2) * r_kj(2) for n >= 3
This formula introduces a new variable, r_kj(2), which represents the two-step transition probability from state k to state j. However, in the original recursion formula, we only have a single-step transition probability, p_kj. Therefore, this formula is not a valid recursion formula for r_ij(n) because it assumes a two-step transition instead of a single-step transition.
3. r_{ij}(n) = sum_(k=1)^(m) sum_(ell=1)^(m) r_ik(n-2) * p_kell * p_ellj for n >= 3
In this formula, we have introduced a double summation, where we sum over two variables, k and ell. The first sum represents the n-2 step transition probability from state i to state k, weighted by the transition probability p_kell. The second sum represents the one-step transition probability from state k to state j, weighted by the transition probability p_ellj. This formula is closer to the original recursion formula, as it considers the n-2 step transition probability and the one-step transition probability. Therefore, this formula is a valid recursion formula for r_ij(n).
In summary:
1. r_{ij}(n) = sum_(k=1)^(m) p_ik * r_kj(n-1) for n >= 2 (valid)
2. r_{ij}(n) = sum_(k=1)^(m) r_ik(n-2) * r_kj(2) for n >= 3 (not valid)
3. r_{ij}(n) = sum_(k=1)^(m) sum_(ell=1)^(m) r_ik(n-2) * p_kell * p_ellj for n >= 3 (valid)