This is a follow up of this post. I.e.:
Suppose that there are two kinds of cells, type $A$ and type $B$.
Let $A_t$ and $B_t$ be the number of cells (i.e. $\mathbb{N}_{0}$-valued) of type $A$ and type $B$, respectively, at time $t\in\mathbb{R}_{+}$.
Each cell of type $A$, independent of all other cells is, after a time that is exponential distribution with parameter $\lambda>0$, split into
- two cells of type $A$ with probability $p_1> 0$,
- two cells of type $B$ with probability $p_2> 0$,and
- one cell of type $A$ and one cell of type $B$ with probability $p_3> 0$,
where $p_1+p_2+p_3 = 1$.
Each cell of type $B$, independent of all other cells, after a time that is exponential distribution with parameter $\gamma>0$, dies. i.e. the number of cells of type $B$ decreases by one.
New questions:
Let $a\in\mathbb{N}$ and $b\in\mathbb{N}_{0}$. What is \begin{align} \mathbb{E}[A_t\mid A_0=a \text{ and } A_s > 0 \text{ for each } s\in[0,t)] \end{align} and \begin{align} \mathbb{E}[B_t\mid A_0=a,B_0=b \text{ and } A_s > 0 \text{ for each } s\in[0,t)] \end{align} for each time $t\in\mathbb{R}_+$? I.e. the expected populations sizes at time $t$, conditioned on non-extinction of cells of type $A$.