 Info, referenser, m.m... μ:

σ:   Mixture of Poisson distributions – an animation

General.   This animation is based on a process at a public general health care center. The data was the number of incoming phone calls per day. It was noted that the intensity of calls was highest on Monday and then decresed over the week until Friday.
A process like 'events per...' is often modelled as a Poisson variable. However, as the intensity varied over the week it was obvious that the day-to-day variation is a mixture of such variables.

The real data.   The data came from the first six months of 2014 and had an average of 87.7 calls per day and a standard deviation of 17.4. When using the five estimated intensities (not shown here) and the formulas shown under the button [Formulas] the result was 87.4 and 16.3. Thus a rather good fit between theory and data.

From the info above it is easy to see that Poisson variables are very common when studying 'counting'-processes.

Note.  The sum (not the mixture) of a number of Poisson variables is a new Poisson distribution. This means that 'The total number of incoming calls per week' is a Poisson variable.

••••

Lambda distr 1

Lambda distr 2

Lambda distr 3

Lambda distr 4

Lambda distr 5

Exercise 1 – change the parameters
Change the parameters via the slides and note that the distribution changes accordingly. If necessary, change the min or max values of the X-axis.

Set all the parameters to 100 and notice that the expected value is 100 and the standard deviation is 10.

In a Poisson distribution the standard deviation equals the square root of the mean.

Notice also that plus minus three sigma includes practically all probability.

(In this animation the Poisson is always nearly symmetrical as the parameters are > 16, a common rule of thumb).

••••

After reading the 'info'-fields and performing the exercises, it is obvious that a mixture of distributions can be difficult to find.
Usually there is a need for other variables that indicate e.g. machine or similar.
If the data consists of a sudden change in mean, this can sometimes be found by e.g. SQC-metods or other types of time series analysis.

••••

The expected value where pi (0 < pi < 1) is the proportion of each Poisson distribution. In this animation there are five Poisson distribution with equal weight, thus each p = 0.2:

The standard deviation:

The pdftot is the 'height' of the mixed distribution at every X-value:

••••

The blue vertical lines are the resulting mixed distribution and each bar it the probability of the corresponding X-value. The sum of all bars is 1.

The expected value is indicated on the X-axis as one red vertical bar with the value attached to it. The small red lines indicate 1, 2, and 3 sigma from the expected value.

The X-axis can be changed by clicking and changing the min or max values for a better fit.

••••

Exercises.  A number of exercises to further illuminate certain features of mixture of variables.

Some conclusions.  A summary of the main ideas and problems with mixture of variables.

Formulas.  There are three main formulas that are used for the mixed result: the expected value, the standard deviation and the probability distribution. These formulas are valid for all distributions.

Change parameters.  It is possible to change the parameters for the mixed distribution. This is done using five sliders.

Mixed Poisson.  The button leads to a page showing a mixture of Poisson distributions.

Mixed normal.  The button leads to a page showing a mixture of normal distributions.

Ordinary Poisson.  The button leads to a page showing all basic features of a Poisson distribution.

Ordinary normal.  The button leads to a page showing all basic features of a normal distribution.

μ.  The theoretical mean of the mixture of distributions.

σ.  The theoretical standard deviation of the mixture of distributions.

••••