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Let $\{X(t)\}_{t>0}$ on $\{0,1,2,3\}$ a birth and death process, with $\lambda(s)=(3-s)^2$ and $\mu(s)=s^2+s$. Assume $P(X(0)=3)=1$ and determine:

(a)$E[X(t)]$;

(b)$Var[X(t)]$.

I don't know how I can start to resolve this exercise. At the beggin I try to resolve the point (a) using the famous example of "a Linear Growth Model with Immigration". That is:

I let $M(t)=E[X(t)]$ and I detrmine M(t) using $M(t+h)=E[X(t+h)]=E[E[X(t+h)|X(t)]]$.

But I think that this reasoning is wrong because I don't know I can use $P(X(0)=3)=1$, that is a condition that the exercise's text give to me.

Can someone explain to me how I can face this exercise? Thank you for your help.

user502400
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  • I don't see how you can answer this question without explicitly finding the transient behaviour of the process (i.e. $\mathbb P(X(t)=i\mid X(0)=3)$ for all $t,i$), which seems to be rather intractable computationally. – Math1000 Jan 01 '23 at 12:50

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