Poisson process: Differential-Difference equations

-

Read Poisson process: Introduction first.

Recall that the Poisson process \(\{N(t)\}\) with parameter \(\lambda\) is a stochastic process characterized by the following probability distribution:
\begin{equation} p_n(t) = \Pr(N(t) = n) = \frac{(\lambda t)^ne^{-\lambda t}}{n!}, ~~ (t \geq 0, n = 0, 1, 2,\cdots) \tag{Eq:PoissonProc}\end{equation}

Differential-Difference Equations

To understand the meaning of a stochastic process, it is often useful to find the differential(-difference) equations satisfied by the probability distribution. 
By differentiating (Eq:PoissonProc) with respect to time \(t\), we have \begin{equation} \frac{dp_0(t)}{dt} = -\lambda p_0(t) \tag{Eq:DD0}\end{equation} and \begin{equation} \frac{dp_n(t)}{dt} = \lambda p_{n-1}(t) - \lambda p_{n}(t) ~ (n \geq 1). \tag{Eq:DDn}\end{equation}
These are the differential-difference equations satisfied by the sequence of probabilities \(\{p_0(t), p_1(t), \cdots \}\). It is "differential" because of the derivatives \(dp_n(t)/dt\), and "difference" because of the sequence of probabilities \(p_0(t), p_1(t), \cdots\).

Exercise. Derive the above differential-difference equations.

So, how do these differential-difference equations help us understand the Poisson process? To see that, let us discretize the equations. Recall the definition of derivatives:
\begin{equation} \frac{dp_n(t)}{dt} = \lim_{\delta t\to 0}\frac{p_n(t + \delta t) - p_n(t)}{\delta t}. \end{equation}
That is, we can approximate the derivative on the left-hand side by the right-hand without the limit. Then (Eq:DD0) and (Eq:DDn) can be approximated as
\begin{equation} p_0(t + \delta t) = (1 - \lambda\delta t)p_0(t)\end{equation} and \begin{equation} p_n(t+\delta) = (\lambda\delta t)p_{n-1}(t) + (1 - \lambda \delta t)p_n(t) \end{equation}
Now, let us define the following conditional probabilities:
$$\begin{eqnarray} \Pr(N(t+\delta t) = n+1| N(t) = n) & = & \lambda \delta t, \\ \Pr(N(t+\delta t) = n | N(t) = n) & = & 1 -\lambda\delta t. \end{eqnarray}$$
That is, given \(N(t) = n\), after the time interval of \(\delta t\), the value of \(N(t+\delta t)\) either increase by 1 (with probability \(\lambda \delta t\)) or stays the same (with probability \(1 - \lambda \delta t\)). We should assume that \(\lambda \delta t \leq 1\) so that it can be interpreted as a probability. Using these conditional probabilities, the discretized differential-difference equations are expressed as
$$\begin{eqnarray} \Pr(N(t + \delta t) = 0) &=& \Pr(N(t+\delta t) = 0 | N(t) = 0)\Pr(N(t)=0),\tag{Eq:DD0'}\\ \Pr(N(t+\delta)=n) &=& \Pr(N(t+\delta t) = n | N(t) = n-1)\Pr(N(t) = n-1) \nonumber\\ &&+ \Pr(N(t+\delta t) = n | N(t) = n)\Pr(N(t)=n). \tag{Eq:DDn'} \end{eqnarray}$$
Note that these are in the form of the partition theorem. From these equations, we can see that the Poisson process is a Markov process because each \(p_n(t+\delta t)\) depends only on the quantities at time \(t\) such as \(p_{n-1}(t)\) and \(p_{n}(t)\), not on those before \(t\) such as  \(p_{m}(t - \delta t), p_{m}(t-2\delta t), p_{m}(t - 3\delta t), \cdots \) for any \(m = 0, 1, \cdots\).
(Eq:DD0') says:
  • \(N(t+\delta t) = 0\) at time \(t + \delta t\) with probability \(p_0(t+\delta t)\) if \(N(t) = 0\) at time \(t\) with probability \(p_0(t)\), and \(N(t)\) stays the same with probability \(1 - \lambda \delta t\).
Similarly, (Eq:DDn') says
  • \(N(t+\delta t) = n\) with probability \(p_n(t + \delta t)\) if either
    • \(N(t) = n-1\) with probability \(p_{n-1}(t)\) and \(N(t)\) increases by 1 with probability \(\lambda \delta t\),
    • or \(N(t) = n\) with probability \(p_n(t)\)and \(N(t)\) stays the same with probability \(1 - \lambda \delta t\). 
Thus, the Poisson process is a process where the value of \(N(t)\) increases by at most 1 within a short time interval \(\delta t\) with probability proportional to the interval: \(\lambda\delta t\).



Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Applications of multiple integrals

-
Image
We can use multiple integrals to compute areas and volumes of various shapes. Area of a planar region Definition (Area) Let \(D\) be a bounded closed region in \(\mathbb{R}^2\). \(D\) is said to have an area if the multiple integral of the constant function 1 over \(D\), \(\iint_Ddxdy\), exists. Its value is denoted by \(\mu(D)\): \[\mu(D) = \iint_Ddxdy.\] Example . Let us calculate the area of the disk \(D = \{(x,y)\mid x^2 + y^2 \leq a^2\}\). Using the polar coordinates, \(x = r\cos\theta, y = r\sin\theta\), \(dxdy = rdrd\theta\), and the ranges of \(r\) and \(\theta\) are \([0, a]\) and \([0, 2\pi]\), respectively. Thus, \[\begin{eqnarray*} \mu(D) &=& \iint_Ddxdy\\ &=&\int_0^a\left(\int_0^{2\pi}rd\theta\right)dr\\ &=&2\pi\int_0^a rdr\\ &=&2\pi\left[\frac{r^2}{2}\right]_0^a = \pi a^2. \end{eqnarray*}\] □ Volume of a solid figure Definition (Volume) Let \(V\) be a solid figure in the \((x,y,z)\) space \(\mathbb{R}^3\). \(V\) is sai
Read more