I have a problem where cars are entering an area according to a homogeneous Poisson process, with a rate of $\lambda = 9$ cars per hour. 20% of the cars entering the area are of type 1, and 80% of the cars entering the area are of type 2. I know that the type of arriving car is independent of the time of arrival, independent of the history of arrivals, and independent of the types of other vehicles to have arrived. I have a Markov process $X(t)$ that denotes the number of cars to arrive within a time interval of $t$ hours starting at 7am tomorrow.
I want to derive $\text{Corr}(X(s), X(t))$, the correlation coefficient of $X(s)$ and $X(t)$, for $0 \le s \le t$. However, when researching this, the only results I find are advanced research papers. How would one derive $\text{Corr}(X(s), X(t))$?
oh, my comment was actually incorrect. sorry. I mixed up covariance and correlation.
\begin{align} \text{corr}(X(s),X(t)) &= \frac{\text{cov}(X(s),X(t))}{\sqrt{\text{var}(X(s))\text{var}(X(t))}}\\ &= \frac{\text{cov}(X(s),X(s))}{\sqrt{\text{var}(X(s))\text{var}(X(t))}}\\ &= \sqrt{\frac{\text{var}(X(s))}{\text{var}(X(t))}}\\ \end{align}
the variance of a poisson distribution you can either just look up on wikipedia, or compute it yourself.