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 $p(t)$?

Given that time $t_0$ has already passed, we need to renormalize the distribution since there is now zero probability that the event occurs before $t_0$. Simply set $p(t)=0$ for $t<t_0$, integrate the remaining curve to find the area, and divide $p(t)$ by that area to get the new probability density. This ensures that the event can only occur between $t_0$ and $T$, while still keeping the area under the probability density function equal to $1$.
