1. Binomial distribution

• Problem: Toss a coin $n$ times, let $X=1$ represents "obtain a head" and $X=0$ represents "obtain a tail" where $\mathbf{Pr}(X=1)=p$. The probability of obtaining $x$ times $X=1$ is:

2. Poisson distribution

• Problem: A store serves $5.1$ customers per hour on average. We want to find the distribution of the number of customers in the specific period of time.

  • We can use binomial distribution, but first at all, we have to scale the number of customers per hour to the range $[0,1]$. For example, we can use:

    • per minute ($n=60$): $p=\displaystyle\frac{5.1}{60}=0.085$
    • per second ($n=3600$): $p=\displaystyle\frac{5.1}{3600}=0.001417$. And now we can use formula $(1)$ to estimate the number of customers per hour/minute/second.

  • But in Poisson distribution, we have a new approach. Consider:

  $$\begin{align} \displaystyle\frac{f(x+1)}{f(x)}&=\displaystyle\frac{\begin{pmatrix}n\\x+1\end{pmatrix}p^{x+1}(1-p)^{n-x-1}}{\begin{pmatrix}n\\x\end{pmatrix}p^x(1-p)^{n-x}}\\ &=\displaystyle\frac{(n-x)p}{(x+1)(1-p)}. \end{align}\tag2$$
  • If we scale $p$ into small periods ($ms$, $\mu s$, ...), then $p\to0^+$ and $n>>x$. So we have some approximations $n-x\approx n$, $1-p\approx1$ and:
  $$\begin{align} \displaystyle\frac{f(x+1)}{f(x)}&\approx\displaystyle\frac{np}{x+1}\\ \Rightarrow f(x+1)&\approx f(x)\displaystyle\frac{\lambda}{x+1}\ \ \ \ \text{where }\lambda=np\\ &=f(0)\displaystyle\frac{\lambda^{x+1}}{(x+1)!}\\ \Rightarrow f(x)&=f(0)\displaystyle\frac{\lambda^{x}}{x!}.\tag3 \end{align}$$
  • Because $f(x)$ is a distribution, we have:
  $$\begin{align} \sum_{x=0}^{+\infty}f(x)&=1\\ \Leftrightarrow f(0)\sum_{x=0}^{+\infty}\frac{\lambda^{x}}{x!}&=1\\ f(0)&=\displaystyle\frac{1}{\sum_{x=0}^{+\infty}\displaystyle\frac{\lambda^{x}}{x!}}.\tag4 \end{align}$$
  • By Taylor approximation, we can obtain $f(0)=\displaystyle\frac{1}{e^\lambda}=e^{\lambda}$.

• Definition: The Poisson distribution of random variable $x$ with the mean $\lambda>0$ takes the form:

• From our approach, we have a approximation between Binomial and Poisson:

for all $x=0,1,...$

Reference:

• Chapter 5.4 | Probability and Statistics (Fourth Edition) | Morris H. DeGroot & Mark J. Schervish.