Poisson Distribution

Math Probability

The Poisson Distribution

The Poisson Distribution describes a distribution where an event occurs for an interval of time, where there is an a mean number of times the event happens in the same interval of time.

Binomial Distribution to Poisson Distribution

Binomial Distribution

$$\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k}$$

Binomial Distribution with infinite trials

$$\underset{n\to \infty }{\lim }\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k}$$

Let a be np, the average success rate in n intervals. This gives us the Poisson Distribution in another form.

$$\underset{n\to \infty }{\lim }\frac{n!}{k!(n-k)!}(\frac{a}{n}{)}^k(1-\frac{a}{n}{)}^{n-k}$$ $$\underset{n\to \infty }{\lim }\frac{n!}{k!(n-k)!}(\frac{a^k}{n^k})(1-\frac{a}{n}{)}^n(1-\frac{a}{n}{)}^{-k}$$ $$\frac{a^k}{k!}\underset{n\to \infty }{\lim }\frac{n!}{n^a(n-k)!}(1-\frac{a}{n}{)}^n(1-\frac{a}{n}{)}^{-k}$$

Now we have three limits to evaluate

Evaluating the Limits

First Limit

$$\underset{n\to \infty }{\lim }\frac{n!}{n^k(n-k)!}$$ $$\underset{n\to \infty }{\lim }\frac{n(n-1)(n-2)...(n-k)(n-k-1)...(1)}{n^k(n-k)(n-k-1)...(1)}$$ $$\underset{n\to \infty }{\lim }\frac{n(n-1)...(n-k+1)}{n^k}$$ $$\underset{n\to \infty }{\lim }(\frac{n}{n})(\frac{n-1}{n})...(\frac{n-k+1}{n})$$

As n goes to infinity, all the terms tend to 1. Therefore, the limit tends to 1.

Second Limit

$$\underset{n\to \infty }{\lim }(1-\frac{a}{n}{)}^n$$

Let u be -n/x (note this tends to negative infinity)

$$\underset{n\to \infty }{\lim }(1+\frac{1}{u}{)}^{-au}$$

Use definition of e

$$e^{-a}$$

Third Limit

$$\underset{n\to \infty }{\lim }(1-\frac{a}{n}{)}^{-k}$$

a/n tends to 0

$$1^k$$

Therefore this limit tends to 1.

Putting it together

$$\frac{e^{-a}{a}^k}{k!}$$

is the formula for the probability of an event happening k times in an interval of time, where a is the mean number of times of the event happening in the interval of time the event ran in. This is the formula for the Poisson Distribution.