Poisson process: Generating function

-

 In a previous post, we solved the differential-difference equations for the Poisson process by using the iterative method. In this post, we use the probability generating function (PGF) to solve the same problem.

Let us restate the problem for the sake of completeness.

We consider a stochastic process \(\{N(t)\}\) that assumes random non-negative integer values \(N(t) = 0, 1, \cdots\) with continuous time \(t \geq 0\). In a previous post, we derived the following differential-difference equations from a few simple assumptions:

\[\begin{eqnarray} \frac{dp_0(t)}{dt} &=& -\lambda p_0(t),\tag{DD0}\\ \frac{dp_n(t)}{dt} &=& \lambda p_{n-1}(t) - \lambda p_n(t), ~~ (n \geq 1).\tag{DDn} \end{eqnarray}\]

We would like to solve these equations with the initial condition \[p_0(0) = 1.\] This implies that \(p_n(0) = 0\) for \(n \geq 1\). We can summarize these initial conditions as \[p_n(0) = \delta_{0,n}, ~~ n\geq 0 \tag{init}\] where \(\delta_{0,n}\) is Kronecker's delta.

In general, the PGF is defined as \[G(s,t) = \sum_{n=0}^{\infty}p_n(t)s^n\tag{Eq:pgf}\] where \(s\) is the dummy variable. Note that this PGF also depends on time \(t\) (because \(p_n(t)\) depends on \(t\)), so it is a two-variable function. 

Now, multiply both sides of Eq. (DDn) by \(s^n\), and sum over \(n = 0, 1, 2, \cdots\). We have \[\sum_{n=0}^{\infty}\frac{dp_{n}(t)}{dt}s^{n} = \lambda\sum_{n=1}^{\infty}p_{n-1}(t)s^{n} - \lambda\sum_{n=0}^{\infty}p_n(t)s^{n}.\]

Let's rewrite this equation using the PGF (Eq:pdf). The left-hand side: \[\sum_{n=0}^{\infty}\frac{dp_{n}(t)}{dt}s^{n} =\frac{\partial}{\partial t}\sum_{n=0}^{\infty}p_{n}(t)s^{n} = \frac{\partial G(s,t)}{\partial t}.\]

The first term on the right-hand side:\[\lambda\sum_{n=1}^{\infty}p_{n-1}(t)s^{n}=  \lambda\sum_{m=0}^{\infty}p_{m}(t)s^{m+1} = sG(s,t).\]

The second term on the right-hand side:\[\lambda\sum_{n=0}^{\infty}p_n(t)s^{n} = G(s,t).\]

After all, we have the following partial differential equation (PDE):

\[\frac{\partial G(s,t)}{\partial t} = \lambda s G(s,t) - \lambda G(s,t) = \lambda(s-1)G(s,t).\]

Although this is a PDE, it only involves a derivative with respect to \(t\). Thus, it can be readily integrated to give 

\[G(s,t) = A(s)e^{\lambda(s-1)t} \tag{Eq:solA}\]

 where \(A(s)\) is a function of \(s\) alone. With the initial condition Eq. (init) and the definition of the PGF (Eq:pdf), we have 

\[G(s,0) = \sum_{n=0}^{\infty}p_n(0)s^n = 1.\] On the other hand, we have from (Eq:solA) 

\[G(s, 0) = A(s).\]

Comparing these, we have \(A(s) = 1\), and the PGF is \[\begin{eqnarray} G(s,t) &=& e^{\lambda(s-1)t} = e^{-\lambda t}e^{\lambda st}\\ &=& e^{-\lambda t}\sum_{n=0}^{\infty}\frac{(\lambda st)^n}{n!}\\ &=& \sum_{n=0}^{\infty}\frac{(\lambda t)^ne^{-\lambda t}}{n!}s^{n}\\ &=& \sum_{n=0}^{\infty}p_n(t)s^{n}. \end{eqnarray}\]

Comparing the coefficients of \(s^n\), we have \[p_n(t) = \frac{(\lambda t)^ne^{-\lambda t}}{n!}, ~~(n = 0, 1, \cdots)\] as expected.


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