## Birth and Death Processes / First-order difference equation question

Hi Everyone,

I have become stuck on the following problem. Here is the statement:

In a simple birth and death process, the birth rates are
$p_{i}=bi$ and the death rates are $q_{i}=di$ for $i=1,\dots,N-1$. The birth and death rates are 0 elsewhere. The parameters $b$ and $d$ are positive and satisfy $(b+d)N\leq1$. Let $\mu_{n}=E[X_{n}]$ . Show that $\mu_{n}$ satisfies the following first-order difference equation:

$\mu_{n+1}=(1+b-d)\mu_{n}+(d-b)N(P(X_{n}=N))$.

I am thinking that I can use some sort of recurrence relation like $E_{y}[X_{n}]=E_{y-1}[X_{n-1}]p_{y-1}+E_{y+1}[X_{n-1}]q_{y+1}$ where $E_{y}[X_{n}]$ denotes the expectation of $X_n$ for a Markov chain starting at y. However, I haven't been able to get very far with this. One thing that is confusing me is how to relate the expectation of a variable in a chain starting in a particular state with the expectation used in the problem statement (which doesn't seem to specify which state the markov chain started in). Do I have to account for all the possible states that the chain might be in?

Any hints would be greatly appreciated--I'm a little frustrated right now.