poisson distribution

The poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event.

Poisson Distribution

A discrete random variable $X$ follows a Poisson distribution with parameter $\lambda >0$, denoted $X\sim \text{Poisson}(\lambda )$, if:

$$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}\text{for }k=0,1,2,\dots$$

where $\lambda$ is the expected number of events in the interval (it’s both the mean and variance of the distribution).

Poisson as limit of Binomial $⟹$ Perfect for modeling rare events in large populations.

The Poisson distribution emerges as the limit of a binomial distribution when the number of trials $n$ becomes large while the expected number of successes $\lambda =np$ remains fixed:

If $X\sim \text{Binomial}(n,p)$ with $\lambda =np$, then as $n\to \infty$:

$$\begin{array}{rl}P(X=k)& =\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}\\ & =\frac{n!}{k!(n-k)!}{\left(\frac{\lambda }{n}\right)}^k{\left(1-\frac{\lambda }{n}\right)}^{n-k}\\ & =\frac{{\lambda }^k}{k!}\underset{\to 1}{\underbrace{\frac{n(n-1)\cdots (n-k+1)}{n^k}}}\underset{\to e^{-\lambda }}{\underbrace{{\left(1-\frac{\lambda }{n}\right)}^n}}\underset{\to 1}{\underbrace{{\left(1-\frac{\lambda }{n}\right)}^{-k}}}\\ & \to \frac{{\lambda }^k}{k!}{e}^{-\lambda }\end{array}$$

Poisson Processes and the exponential distribution

If events occur according to a Poisson process with rate $\lambda$ (events per unit time):
Waiting times between consecutive events are $w_i\sim \text{Exponential}(\lambda )$ (independent)
Number of events in any interval of length $t$ is $N(t)\sim \text{Poisson}(\lambda t)$
This means $P(\text{no events in next }t\text{ time})=P(N(t)=0)=e^{-\lambda t}$, which equals $P(T>t)=e^{-\lambda t}$ where $T$ is the exponential waiting time!

Email arrivals at rate 5/min

If emails arrive at rate $\lambda =5$ per minute:

• Exponential perspective: Time until next email has $P(T>t)=e^{-5t}$
• Poisson perspective: Number of emails in $t$ minutes has $P(k\text{ emails})=\frac{(5t{)}^k{e}^{-5t}}{k!}$

For example:

• Prob. no emails in next minute: $P(N(1)=0)=e^{-5}\approx 0.0067$
• Prob. exactly 1 email in next minute: $P(N(1)=1)=5e^{-5}\approx 0.034$