15 Lecture 14, Feburary 07, 2024
15.1 Poisson distribution
Definition 15.1 (Poisson distribution) We say the random variable \(X\) has a {} distribution with parameter \(\mu > 0\) if \[ f(x) = e^{-\mu} \frac{ \mu^x}{x!},\;\;x=0,1,2,\dots\]
15.2 Poisson as the limiting distribution of the binomial distribution
One way to view the Poisson distribution is to consider the limiting case of binomial distribution, where you fix \(\mu = np\) , and let \(n \rightarrow \infty\) and \(p \rightarrow 0\).
One can show that if \(n\to \infty\) and \(p=p_n \to 0\) as \(n\to \infty\) in such a way that \(n p_n \to \mu\), then \[ {n \choose x} p^x (1-p)^{n-x} \to e^{-\mu} \frac{ \mu^x}{x!},\;\;\;as\;\;\; n\to \infty. \] Actually here, it is something called the *convergence in distribution**.
15.3 Poisson process
Consider counting the number of occurrences of an event that happens at random points in time (or space). Poisson process is the counting process that satisfies the following
Independence: the number of occurrences in non-overlapping intervals are independent.
Individuality: for sufficiently short time periods of length \(\Delta t,\) the probability of 2 or more events occurring in the interval is close to zero \[ \frac{P\left( \text{2 or more events in }(t,t+\Delta_t)\right)}{\Delta_t} \rightarrow 0,\;\; \Delta_t \to 0 \]
Homogeneity or Uniformity: events occur at a uniform or homogeneous rate \(\lambda\) and proportional to time interval \(\Delta_t\), i.e. \[ \frac{P\left( \text{one event in }(t,t+\Delta_t)\right) - \lambda\Delta_t }{\Delta_t} \to 0. \]
If \(X=\) occurrences in a time period of length \(t\), then \[X\sim Poi(\lambda t).\]
Definition 15.2 (Poisson process) A process that satisfies the prior conditions on the occurrence of events is often called a Poisson process. More precisely, if \(X_t, \; \text{for } t\ge0,\) (a random variable for each \(t\)) denotes the number of events that have occurred up to time \(t\), then \(X_t\) is called a Poisson process.
15.4 Side notes – Rigorous definition of convergence in distribution
This section is just served as a reference for those of you who are interested in the rigorous definition of convergence in distribution. Do not worry too much if you are not interested in knowing those.
Definition 15.3 (Convergence in distribution) Let \((F_n)_{n\in\mathbb{N}}\) and \(G\) be CDFs. Let \(c(G) = \{x\in\mathbb{R} : G \text{ is cts. at }x\}\) be the set of continuity points of \(G\). \(F_n\) converges in distribution to \(G\) if \[\begin{align} \forall x\in c(G) \quad F_n(x)\to G(x) \tag{15.1} \end{align}\] If \(X_n\) has CDF \(F_n\) for each \(n\) and \(Y\) has CDF \(G\) and (15.1) holds then \(X_n\) converges in distribution to \(Y\). Denoted \(X_n \stackrel{d}{\to} Y\), or \(F_n \stackrel{d}{\to} G\)