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

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 :)

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 $$dt$$.
• Usually, if $T$ is the waiting time of an event, or the lifetime of something, etc, then $F(t):=P(T\leq t)$ is the CDF, the probability of the event arriving before time $t$ (right? The event $\{T\leq t\}$ denotes when the waiting time is less than $t$ so it must have occurred before time $t$). It is the so-called survival function $G(t)=1-F(t)=P(T>t)$ which denotes the probability of the event arriving after time $t$. Further for the continuous case, $F’(t)=f(t)$ is the PDF, the values of which are not probabilities at all (it is called the probability density function for a reason). – Nap D. Lover Oct 16 '18 at 1:26