Here is the question.
Consider a pure birth process $\{X(t) : t ≥ 0\}$ with birth parameters $\lambda_{2n} = α>0$ and $\lambda_{2n+1} =β>0$ for $n∈N$. Compute $Pr\{X(t) \text{ is odd } \mid X(0)=0\}$ for $t≥0$.
I am aware that the transitional probability equations for a pure birth process are: $$ P_{ii}(t) = e^{-\lambda_{i} t} $$ $$ P_{ij}(t) = \lambda_{j-1} e^{-\lambda_{j} t} \int_{0}^{t} e^{\lambda_{j}s}P_{i,(j-1)}(s)ds, \quad (j > i) $$
I'm pretty sure the transitional probability equations i listed above are what I have to use to start this problem, but I am not sure how to use them. Any insight to this would be greatly appreciated.