2007 FRM Learning outcome 2.7: Calculate the expected value and variance of the Poisson distribution.

What is the Poisson used for?

The Poisson is related to the binomial distribution. In the binomial distribution, we conduct Bernoulli trials. That is, trials with two outcomes; e.g., success/failure, up/down, incoming service center call/no call. We use the Poisson to characterize variables over time (or over space). For example,

• http://www.bionicturtle.com/assets/images/article-li.gif); background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: 4px 6px; background-repeat: no-repeat no-repeat; border: 0px initial initial;">How many operational errors are expected per day/month/year?
• http://www.bionicturtle.com/assets/images/article-li.gif); background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: 4px 6px; background-repeat: no-repeat no-repeat; border: 0px initial initial;">How many typos are expected per page?

What is the pdf?

Hopefully FRM candidates know by now that pdf refers to probability density function (pdf). Note the pdf is also sometimes called the probability mass function. We can describe a distribution by its local density (what is probability that random variable X will equal value x?) or its cumulative distribution function (what is the probability that random variable X will be less than or equal tovalue x?). Further, distributions can be discrete or continuous (e.g., the normal).

The Poisson is discrete:

Lambda is the key parameter. Lambda is both the mean (expected value) and the variance! What is the expected value of an an event over time? It is simply the rate of occurrence (r) multiplied by time (t). So we could expand the Poisson pdf out to this:

So, the things to remember are:

• http://www.bionicturtle.com/assets/images/article-li.gif); background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: 4px 6px; background-repeat: no-repeat no-repeat; border: 0px initial initial;">Poisson is discrete
• http://www.bionicturtle.com/assets/images/article-li.gif); background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: 4px 6px; background-repeat: no-repeat no-repeat; border: 0px initial initial;">The expected value is the mean (rate x time) which is called lambda. Lambda is both the mean and the variance

An example

Let's say we observe our company commits a certain sort of operational error (e.g., bad invoice) ninety times per month. How many errors should we expect in a single day. The mean is (90)(1/30) = 3. So, lambda is three and three is the expected number of errors per day. The Poisson distribution is given below in the EditGrid spreadsheet. Note that I used both the built-in formula and the actual function. You can open your own read/write copy here.