I have a continuous time-dependent point process $\mathcal{X}({t})$ where $t\in[0,1]$ and I am trying to compute the distribution of the occurrence of the first point. The information I have is
$$P(\text{a point occuring at time t}| \text{no point before time t})=\frac{t^3}{3}$$
I am looking to compute
$$P(\text{the first point occurs before or at t})=1-P(\text{no point occurs up to and including time $t$})$$
This information should be all I need to calculate this distribution, but I am unsure how to compute it. Would I have to integrate over $P(\text{a point occuring at time t}| \text{no point before time t})$ from $0$ to $t$?