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Insights

All You need is µ!

Part 2 of Statistics within Capacity Management and Performance Engineering Series.

Last week we looked at how the Poisson limit allows you to take the equation for Binomial Distribution and manipulate it (under the conditions specified) so that all you need is the population mean; a statement which to us is much more useful and can be applied in much wider cases. Herein a proof is given to show that all you need is µ.

We know that from last week the equation for Poisson Distribution is:

(1) formula_b1

We also know by definition that:

(2) formula_b2

From this, putting (1) into (2) we can derive the following:

(3) formula_b3

For reference in this step I have taken µ out as a factor as well as e and cancelled down.

(4) formula_b4

Let:

(5) formula_b5

This gives:

(6) formula_b6

And since

(7) formula_b7

Now we can substitute back into (6)

(8) formula_b8

For extra reference, the standard deviation:

(9) formula_b9

Hence only µ is needed to determine the Poisson distribution.

What we have shown here is logically what was possibly expected; however from a purely mathematical point of view this succinctly shows how truly powerful the Poisson Distribution can be. As a tool used within an analytical model, it is something that has great potential, don’t overlook what it can show.

Next time: Foundations in Linear Approximations