Poisson process: Derivation from simple assumptions

-

 In a previous post, we have seen that the Poisson process satisfies a set of differential-difference equations. By reversing that argument, we can "derive" the Poisson process from a few simple assumptions. 

Let's pretend we don't know the Poisson process. We want to model the stochastic process \(\{N(t)\}\) that assumes non-negative integer values \(N(t) = 0, 1, 2, \cdots\) and depends on non-negative continuous time \(t \in [0, \infty)\). Our first assumption is the following:

(1) The probability that \(N(t)\) increases by 1 within the time interval of \(\delta t\) is approximately proportional to \(\delta t\). In other words,

\[\Pr(N(t + \delta t) = n + 1 | N(t) = n) = \lambda \delta t + o(\delta t)\]

where \(\lambda\) is a positive constant. "\(o(\delta t\)" in the "little o" notation that represents any term that converges to 0 faster than \(\delta t\) when \(\delta t \to 0\). That is,

\[\lim_{\delta t\to 0}\frac{o(\delta t)}{\delta t} = 0.\]

Our next assumption is:

(2) For a sufficiently small \(\delta t\), \(N(t+\delta t) - N(t)\) is no more than 1. That is, \(N(t)\) increases by 1, or it doesn't increase at all within \(\delta t\). Accordingly, we have

\[\Pr(N(t + \delta t) = n | N(t) = n) = 1 - \Pr(N(t+\delta t) = n+1| N(t) = n) = 1 - \lambda \delta t + o(\delta t).\]

Note that we are implicitly assuming that \(\{N(t)\}\) is a Markov process.

Let \(p_n(t) = \Pr(N(t) = n)\) denote the probability that \(N(t) = n\) at time \(t\). 

By the above assumptions, we have

\[p_0(t + \delta t) = (1 - \lambda \delta t + o(\delta t))p_0(t),\]

and, for \(n \geq 1\),

\[p_n(t+\delta t) = (\lambda \delta t + o(\delta t))p_{n-1}(t) + (1 - \lambda \delta t + o(\delta t))p_n(t).\]

By rearranging these, we have

\[\begin{eqnarray} \frac{p_0(t + \delta t) - p_0(t)}{\delta t} &=& -(\lambda + o(\delta t)/\delta t)p_0(t),\\ \frac{p_n(t + \delta t) - p_n(t)}{\delta t} &=& (\lambda + o(\delta t)/\delta t)p_{n-1}(t) - (\lambda + o(\delta t)/\delta t)p_n(t). \end{eqnarray}\]

With the limit \(\delta t \to 0\), we have the (familiar) differential-difference equations:

\[\begin{eqnarray} \frac{dp_0(t)}{dt} &=& -\lambda p_0(t),\\ \frac{dp_n(t)}{dt} &=& \lambda p_{n-1}(t) - \lambda p_n(t) ~~ (n \geq 1). \end{eqnarray}\]

By solving these equations with the initial condition \(p_0(0) = 1\), we can show that

\[p_n(t) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}.\]

That is, \(N(t)\) is a Poisson process. We will see how to solve the above differential-difference equations in the following posts.

Related videos



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