How do you recalculate the probability density function if some time has passed and the event has yet to occur? Suppose I have a probability density function for an event that is certain to occur between $0$ and time $T$, $p(t)$.
However, some time $t_0$ has passed and the event has yet to occur.
So I would like to recalculate a normalized pdf that incorporates this information.
I think I am looking for the posterior distribution, something like
$p(t|t>t_0)$
How would I do this for any arbitrary $p(t)$?

Indeed, we search the Bayesian posterior
$$p(t|t>t_0) = \frac{p(t,t>t_0)}{p(t>t_0)} = \frac{p(t)1_{t>t_0}}{p(t>t_0)}.$$
Explanation: The density $p(t)$ is multiplied by $1_{t>t_0}$, to remove the probability for all events $t\leq t_0$ that we know have not happened. The remaining probability density $p(t)1_{t>t_0}$ integrates to $\int p(t)1_{t>t_0} \mathrm{d}t = p(t>t_0)$ since that was the original probability of our observation $t>t_0$. To turn it into a probability density that integrates to $1$, it is normalized by dividing it by $p(t>t_0)$, so that $\int \frac{p(t)1_{t>t_0}}{p(t>t_0)}\mathrm{d}t = \frac{\int p(t)1_{t>t_0}\mathrm{d}t}{p(t>t_0)} = \frac{p(t>t_0)}{p(t>t_0)} = 1$ as required.
