Homework Help - Markov Chains

[spacecataz]

Homework Statement

Suppose X is a Poisson random variable with mean 5. Suppose Zn = nX + 3 for n = 0, 1, 2, . . . . (a) Does Z0 , Z1 , . . . have the Markov property? (b) If Z0 , Z1 , . . . has the Markov property, is it time-homogeneous?

Suppose X and Y are independent Poisson random variable both having mean 5. Suppose Zn = nX + Y for n = 0, 1, 2, . . . . (a) Does Z0 , Z1 , . . . have the Markov property? (b) If Z0 , Z1 , . . . has the Markov property, is it time-homogeneous?

Homework Equations

pmf of Poisson distribution: ($$\lambda^{k}*e^{-\lambda}$$)/k!

The Attempt at a Solution

I'm not really sure where to start with this. I'm inclined to say that for the first part it is a Markov chain but I'm not sure if I need to use the pmf of the Poisson distribution.

I'm also inclined to say that the second part is not a Markov chain because of Y, but I'm really not sure.

Any resources or guidance with this would be greatly appreciated.

Thanks!

 

[mighty2000]

Hi there,

To determine whether a sequence has the Markov property, we need to check if the future state only depends on the current state and not the past states. In the first part, we have Zn = nX + 3, which means that the future state Z_{n+1} only depends on the current state Z_n and not any past states. Therefore, Z0, Z1, ... has the Markov property.

For the second part, we have Zn = nX + Y. In this case, the future state Z_{n+1} depends not only on the current state Z_n, but also on the past state Y. Therefore, Z0, Z1, ... does not have the Markov property.

Regarding time-homogeneity, a Markov chain is said to be time-homogeneous if the transition probabilities remain the same over time. In the first part, since Zn = nX + 3, the transition probabilities do not change over time and hence, it is time-homogeneous. However, in the second part, the transition probabilities depend on the past state Y, which can change over time, making it not time-homogeneous.

I hope this helps! Let me know if you have any further questions.