3.2 Poisson distribution

While the Binomial distribution deals with the likelihood of a particular combination of results from a Bernoulli process, Poisson distribution deals with the probability of how often events occur. For example, knowing that a bus usually arrives at a station every 15 minutes, what is the likelihood (probability) that only three busses arrives in 120 minutes?

The probability distribution function requires two pieces of information. $k$ is usually used to denote ``how many events'', and $\lambda$ is usually used to indicate the expected number of occurances. Thus, in our example, $k=3$ because we are interested in the probability that only three busses arrive. However, $\lambda=120/15=8$ because on the average, a bus arrives every 15 minutes, so there should have been 8 busses over 120 minutes.

In order to apply Poisson probability distribution, we need to assume that the probability of the first event happening after a period of $t$ is $e^{-\lambda}$ , where $\lambda$ is the expected number of occurances in time $t$ .

The random variable (function) $X$ is usually defined as the number of occurances.

The probability function is as follows:

$${\rm Pr}(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$$ (3)

In many cases, we are more interested in the cumulative probability. In other words, we want to know ``what are the chances that on the average, a bus is late in two hours''?

This probability can be expressed as follows:

$${\rm Pr}(X < 8) = \sum_{i=0}^{7} {\rm Pr}(X = i) = \sum_{i=0}^{7} \frac{e^{-8}8^{i}}{i!}$$