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We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-\lambda \Delta t)(\lambda \Delta t)^k}{k!}$.

And therefore, event count in $[0, T]$ follows a Poisson distribution with rate $\lambda T$. That is, $P\left \{ N(T)=k\right \}=\frac{\text{exp}(-\lambda T)(\lambda T)^k}{k!}$. ($N$ is the count.)

The problem is:

Prove that the following simulation generates a homogeneous Poisson process with rate $\lambda$ on $[0, T]$: Step 1: Sample $m$ from Poisson distribution with mean $\lambda T$. Step 2: Sample $s_1, \cdots,s_m$ i.i.d. from uniform $[0, T]$. That is, demonstrate that for any time interval $[t, t+\Delta t]$ in $[0,T]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-\lambda \Delta t)(\lambda \Delta t)^k}{k!}$.

Now we look at the problem, we have

Given $m$ events in $[0,T]$,

$P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}\\ =\Sigma ^{\infty }_{m=k}P\left \{ k \;\text{events in}\; [t, t+\Delta t],m \;\text{events in}\; [0,T]\right \}\\ =\Sigma ^{\infty }_{m=k}P\left \{ k \;\text{events in}\; [t, t+\Delta t] | m \;\text{events in}\; [0,T]\right \}\cdot P\left \{ m \;\text{events in}\; [0,T] \right \}\\ =\sum_{m=k}^{\infty }\binom{m}{k}(\frac{\Delta t}{T})^k(\frac{T-\Delta t}{T})^{m-k} \cdot \frac{\text{exp}(-\lambda T)(\lambda T)^m}{m!}$

So in order to prove the result, we should have

$\sum_{m=k}^{\infty }\binom{m}{k}(\frac{\Delta t}{T})^k(\frac{T-\Delta t}{T})^{m-k} \cdot \frac{\text{exp}(-\lambda T)(\lambda T)^m}{m!}=\frac{\text{exp}(-\lambda \Delta t)(\lambda \Delta t)^k}{k!}$ $(*)$

and this should hold. But my question is how to derive $(*)$ mathematically? How to show the two sides are equal in $(*)$? Can you show it?

Thanks in advance.

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  • $\begingroup$ Apply the identity $$\sum_{m=k}^\infty{m\choose k}a^{m-k}\frac{b^m}{m!}=\frac{b^k}{k!}\sum_{m=k}^\infty\frac{(ab)^{m-k}}{(m-k)!}=\frac{b^k}{k!}\mathrm e^{ab},$$ for the values $$a=\frac{T-\Delta t}T,\qquad b=\lambda T.$$ $\endgroup$ – Did Feb 4 '15 at 10:50

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