Relation between waiting time distribution and probability that an event occurs within time $dt$

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Waiting time distribution is defined as the distribution of the time interval between two successive events. I'm looking at stochastic processes in discrete space and continuous time with non-exponential waiting time distributions. I want to see how these waiting time distributions relate with the probability of an event occurring in time interval $dt$.

While I know for exponential distributions how this can be done, I'm having trouble figuring it out for an arbitrary waiting time distributions.

Any help is appreciated :)

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Oh so waiting time distribution can be thought of as a cumulative distribution (an event not occuring until time $t$) and hence the time derivative should give me the probability distribution of an event not occuring in time interval $dt$.

Sorry about this post.