Link between Poisson distribution and Exponential distribution

Let $(\sigma_i)_{i \geq 1}$ be i.i.d. random variables with Exponential distribution of parameter $\lambda$, representing the waiting time between consecutive events. The arrival time of these events (from a $t=0$ origin) is:

$$\tau_n = \sigma_1 + \sigma_2 + ... + \sigma_n$$

The number of those events happening between time $t_0=k$ and $t_1=k+1$ is:

$$M_k :=\sum_{\quad j \geq 1 \\ k \leq \tau_j < k+1} 1$$

Is it true that $(M_k)$, counting the number of events happening in a time-window $[k, k+1[$, has a Poisson distribution? It seems true intuitively, but I would like to find a source / proof.

NB:

• I have aleady read Link between Poisson and Exponential distribution and Relationship between Poisson and exponential distribution which may be linked, but it's not the same question.

• I wonder if the reciprocal is also true: let's say we have consecutive events with arrival times of $(\tau_i)_{i \geq 1}$, such that the number $M_k$ of events happening between $[k, k+1]$ is a Poisson distribution of parameter $\lambda$, for each integer $k$. Can we conclude that $\sigma_i = \tau_i - \tau_{i-1}$ has an exponential distribution?

Solution 1:

Let $(\sigma_i)$ be an i.i.d. sequence of exponentially distributed waiting times: $$ \mathbb P(\sigma_i\in[t,t+dt])=\lambda\,e^{-\lambda t}\,dt $$ and define $\tau_n:=\sum_{i=1}^n\sigma_i$ to be the time of the $n$-th jump. The jump process is defined as $$ N_t=\sum_{n=1}^\infty 1_{\{\tau_n\le t\}}\,. $$ The distribution of $N_t$ is obtained from \begin{align} &\mathbb P(N_t= k)\\&=\mathbb P\left(\tau_k\le t<\tau_{k+1} \right)=\mathbb P\left(\sigma_1+...+\sigma_k\le t<\sigma_1+...+\sigma_{k+1} \right)\\ &=\mathbb P\Big(\sigma_1\le t,\sigma_2\le t-\sigma_1,...,\sigma_k\le t-\sigma_1-...-\sigma_{k-1},t-\sigma_1-...-\sigma_{k} <\sigma_{k+1} \Big)\\ \tag{1} &=\int_0^t\int_0^{t-s_1}...\int_0^{t-s_1-...-s_{k-1}}\int_{t-s_1-...-s_k}^\infty\lambda^{k+1}e^{-\lambda s_1-...-\lambda s_{k+1}}\,ds_{k+1}\,...ds_1\,. \end{align} The $ds_{k+1}$-integral is \begin{align} \lambda^ke^{-\lambda s_1-...-\lambda s_k}\,e^{-\lambda t+\lambda s_1+...+\lambda s_k} =\lambda^ke^{-\lambda t}\,. \end{align} The $ds_k$-integral is \begin{align} \lambda^k e^{-\lambda t}\big(t-s_1-...-s_{k-1}\big)\,. \end{align} The $ds_{k-1}$-integral is $$ \lambda^k e^{-\lambda t}\big(t-s_1-...-s_{k-2}\big)^2-\lambda^k e^{-\lambda t}\frac{(t-s_1-...-s_{k-2})^2}{2}=\lambda^k e^{-\lambda t}\frac{(t-s_1-...-s_{k-2})^2}{2}\,. $$ And so forth. This shows that (1) is $$ \mathbb P(N_t=k)=\lambda^ke^{-\lambda t}\frac{t^k}{k!}\, $$ which is the well-known Poisson distribution with parameter $\lambda\, t$.

Conversely, when we start with a Poisson process $N_t$ then its first jump time has an exponential distribution function: \begin{align} \mathbb P(\tau_1\le t)=\mathbb P(N_t\ge 1)=\sum_{k=1}^\infty\frac{(\lambda t)^k}{k!}\,e^{-\lambda t} =\Big(\underbrace{\sum_{k=0}^\infty\frac{(\lambda t)^k}{k!}}_{\displaystyle e^{\lambda t}}-1\Big)\,e^{-\lambda t}=1-e^{-\lambda t}\,. \end{align} To find the distribution of $\sigma_2=\tau_2-\tau_1$ we look at the joint distribution of $\tau_1$ and $\tau_2$ first: for $t_1\le t_2\,,$ \begin{align} &\mathbb P(t_1<\tau_1,\tau_2\le t_2)=\mathbb P(N_{t_1}<1,N_{t_2}\ge 2)=\mathbb P(N_{t_1}=0)-\mathbb P(N_{t_1}=0,N_{t_2}\le 1)\\\tag{2} &\stackrel{(*)}{=}e^{-\lambda t_1}-e^{-\lambda t_1}\big(1+\lambda(t_2-t_1)\big)\,e^{-\lambda (t_2-t_1)}=e^{-\lambda t_1}-e^{-\lambda t_2}-\lambda(t_2-t_1)\,e^{-\lambda t_2}\,. \end{align} In (*) we have used the independence of $N_{t_2}-N_{t_1}$ and $N_{t_1}\,.$ Therefore, \begin{align} &\mathbb P(\tau_1\le t_1,\tau_2\le t_2)=\mathbb P(\tau_2\le t_2)- \mathbb P(t_1<\tau_1,\tau_2\le t_2)\\ &\stackrel{(2)}{=} \mathbb P(N_{t_2}\ge 2) -e^{-\lambda t_1}+e^{-\lambda t_2}+\lambda(t_2-t_1)\,e^{-\lambda t_2}\\ &=\sum_{k=2}^\infty\frac{(\lambda t_2)^k}{k!}\,e^{-\lambda t_2}-e^{-\lambda t_1}+e^{-\lambda t_2}+\lambda(t_2-t_1)\,e^{-\lambda t_2}\\&=1-e^{-\lambda t_2}-\lambda t_2\,e^{-\lambda t_2}-e^{-\lambda t_1}+e^{-\lambda t_2}+\lambda(t_2-t_1)\,e^{-\lambda t_2}\\ &=1-\lambda t_2\,e^{-\lambda t_2}-e^{-\lambda t_1}+\lambda(t_2-t_1)\,e^{-\lambda t_2}\,. \end{align} Consequently, $$\tag{3} \mathbb P\Big(\tau_1\in[t_1,t_1+dt],\tau_2\in[t_2,t_2+dt]\Big)=\lambda^2e^{-\lambda t_2}1_{\{t_1\le t_2\}}\,dt_1\,dt_2\,. $$ It follows that $\sigma_2$ has an exponential distribution: \begin{align} &\mathbb P(\sigma_2\le t)=\mathbb P(\tau_2\le t+\tau_1)=\int_0^\infty\int_{t_1}^{t+t_1}\lambda^2e^{-\lambda t_2} \,dt_2\,dt_1=\lambda \int_0^\infty e^{-\lambda t_1}-e^{-\lambda(t+t_1)}\,dt_1\\ &=1-e^{-\lambda t}\,. \end{align} The remaining task is now to show this for all $\sigma_n$. I suspect that the joint distribution of $\tau_{n-1}$ and $\tau_n$ is also given by (3).

Solution 2:

It is true and concerns a homogeneous Poisson point process.

More generally if $A\subseteq[0,\infty)$ is a Borel-measurable set with finite Lebesgue measure and: $$N_A=\sum_i1_A(X_i)$$ then $N_A$ has Poisson distribution with parameter $\lambda m(A)$ where $m$ denotes the Lebesgue measure.

Also have a look here.

It can be further generalized in the sense that also other measures can be used (non-homogeneous).

For a homogeneous Poisson point process we need exactly what you describe in your question: a sequence of iid rv's having exponential distribution.

I suspect that the conditions described in your question concerning the reciprocal are not sufficient.

Not only the number of arrivals on intervals $[k,k+1)$ (where $k$ denotes a nonnegative integer) must have Poisson distribution, but also the number of arrivals on other intervals.