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A Poisson process is defined by the probability of encountering $k$ events within an arbitrary time interval $\Delta t$, namely

$P\{N(\Delta t) = k\} = \frac{e^{−λ\Delta t}(λ\Delta t)^k}{k!}$

Further, it can be shown (for example, here), that the time interval between two consecutive events $t_i$ and $t_{i+1}$ follows an exponential distribution. Its probability density is given by

$\rho(t_{i+1} - t_i = T) = \lambda e^{-\lambda T}, \;\forall i$

I am interested in a slightly more complicated distribution for event interval. In my application, the interval follows the exponential distribution, except it can't be less than come constant waiting time $T_0$. Using the Heaviside step function $H(x) = \delta(x > 0)$, I define the interval time probability density as

$\rho(t_{i+1} - t_i = T) = \lambda e^{-\lambda (T-T_0)} H(T-T_0), \;\forall i$

which means it is zero for any interval $T < T_0$, and follows the exponential distribution afterwards.

Question: Calculate the probability of observing $k$ events within a time interval $\Delta t$. If it helps, you may assume that the beginning of the interval is not affected by compulsory interval due to events that may have happened before the interval.

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  • $\begingroup$ You're confusing probability densities with probabilities. In your second displayed equation, the right-hand side is the probability density of the exponential distribution, but the left-hand side is written as if it were a probability. You're also missing a normalization constant in the last displayed equation; the density isn't normalized to $1$ as written. $\endgroup$ – joriki Aug 28 '18 at 9:16
  • $\begingroup$ You're right, the expression for interval is a probability density. I will rewrite the notation in a bit. But I don't see your argument about normalization, I think the last equation integrates to 1 if you integrate w.r.t $T$ from 0 to infinity $\endgroup$ – Aleksejs Fomins Aug 28 '18 at 9:27
  • $\begingroup$ Sorry, you're right about the normalization. $\endgroup$ – joriki Aug 28 '18 at 9:35

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