# Conditioning and a sum of Poisson random variables

Posted by Jason Polak on 13. February 2018 · Write a comment · Categories: statistics · Tags:

Previously we talked about the Poisson distribution. The Poisson distribution with mean $\mu \gt 0$ is a distribution on the natural numbers whose density function is
$$f(n) = \frac{e^{-\mu}\mu^n}{n!}$$
We have already seen that the Poisson distribution essentially arises from the binomial distribution as a sort of "limiting case". In fact, the Poisson distribution is sometimes used as an approximation to the binomial distribution, even though it also arises in its own right in processes like observing the arrival of random particles from radioactive decay.

The Poisson distribution and the binomial distribution are related in another way, through conditioning on the sum of Poisson random variables.

Suppose we have two independent Poisson random variables $X_1$ and $X_2$ with means $E(X_i) = \mu_i$. Then the sum $X_1 + X_2$ also has a Poisson distribution with mean $\mu_1 + \mu_2$.

On the other hand, what is the conditional density $P(X_1 = n_1, X_2 = n_2~|~ X_1 + X_2 = n)$? Here, $n_1 + n_2 = n$. By definition, it is
$$\frac{P(X_1 = n_1, X_2 = n- n_1)}{P(X_1 + X_2 = n)}$$
This is
$$\binom{n}{n_1}p^{n_1}(1-p)^{n-n_1}$$
where $p = \mu_1/(\mu_1+\mu_2)$. So, the joint density of two Poisson random variables conditioning on their sum being $n$ is binomial!

# Where does the Poisson distribution come from?

Posted by Jason Polak on 08. February 2018 · Write a comment · Categories: statistics · Tags: ,

The Poisson distribution is a discrete probability distribution on the natural numbers $0,1,2,\dots$. Its density function depends on one parameter $\mu$ and is given by
$$d(n) = \frac{e^{-\mu}\mu^n}{n!}$$
Not surprisingly, the parameter $\mu$ is the mean, which follows from the exponential series
$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$
Here is what the density function looks like when $\mu=5$:

How does the Poisson distribution actually arise?

It comes from the following process: suppose you have a fixed interval of time, and you observe the number of occurrences of some phenomenon. In practice, it might be 'the number of buses to arrive at a given bus stop'. Whatever it is, you're counting something.

Moreover, this process has to satisfy the important "Poisson axiom": if you take two disjoint intervals of time that are small, then the number of occurrences in the first is independent of the number of occurrences in the second. Here, "small" means that as the size of the intervals approaches zero, the results should approach independence.
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