I'm having a bit of trouble figuring out a problem presented in lecture. Given a pure birth process ${X_t}$ with $X_0 = 0$ and rates $\lambda_i, i\geq0$, we're supposed to figure out for some integer $K$, $P(X_T \geq K)$, where $T$ is an exponential random variable with parameter $\theta$, and is independent of $X_t$.
Right now my thought process is that I'm supposed to do something along the lines of $1 - P(X_t < K | T = t) * P(T = t)$. However, I'm not sure how to compute $P(X_t < K)$ (summation over all values less than $K$? That seems like it would be pretty messy given $X_t$'s pdf and there's probably something much simpler). My other idea involved the interarrival times of $X_t$, but I'm still not exactly sure where to start there. If I'm on the complete wrong track or I completely messed something up I'd appreciate a push in the right direction. Thanks!