Consider a pure birth process starting from $X(0) = 0$ with birth parameters $\lambda_0 = 1.4$ and $\lambda_1 = 1.8$ Compute the following probabilities $\mathbb{P}(X(0.2) = 0)$ and $\mathbb{P}(X(0.2) = 1)$
So far I have calculated $\mathbb{P}(X(0.2) = 0) = \frac{(1.4*0.2)^0*e^{-1.4*0.2}}{0!} = 0.75568 $
But I am having trouble figuring out how to calculate the next one, so far I have tried assuming since we have not reached 1 yet since the time interval is [0,0.2] so we have $(1.4*0.2)^1*e^{-1.4*0.2}$ but it isn't correct and also $(1.8*0.2)^1*e^{-1.8*0.2}$ to no avail. I would appreciate any help with this question
For the second one, you have to consider all possible times at which the birth happened, and it could have happened anywhere in the interval $[0,T]$.
$$ \begin{split}P(X(T)=1)&=\int_0^T dt_1P(X(T)=1|X(t_1)=1)P(X(t_1)=1|X(0)=0)\\&= \int_0^T dt_1e^{-\lambda_1(T-t_1)}(1-e^{-\lambda_0t_1})\\&= \frac{1-e^{-\lambda_1T}}{\lambda_1}+\frac{e^{-\lambda_1T}- e^{-\lambda_0T}}{\lambda_1-\lambda_0} \end{split}$$