Proving the poisson distribution is normalized

1. Oct 3, 2011

Eight

SOLVED

1. The problem statement, all variables and given/known data

I am trying to prove that the poisson distribution is normalized, I think I've got an ok start but just having trouble with the next step.

2. Relevant equations

A counting experiment where the probability of observing n events (0≤n<∞) is given by:

P(n) = (μ^n)/n! * e^(-μ)

Where μ is a real number.

3. The attempt at a solution

Background (possibly incorrect)

So it's discrete, as n will take integer values; I need a sum not an integral.

Ʃ P(n) from n=0 to ∞ is just given by:

Ʃ (μ^n)/n! * e^(-μ)

And e^(-μ) does not vary with n, so:

Ʃ P(n) = e^(-μ) * Ʃ (μ^n)/n!

Important bit

Now as I am trying to prove it is normalized, i need to get Ʃ P(n) = 1, so I assume my problem is getting from:

(μ^n) / n!

to

e^μ

Any tips or help would be much appreciated, thanks in advance.

Last edited: Oct 3, 2011
2. Oct 3, 2011

Ray Vickson

You are doing probability applications but have never seen this material before? Oh, well: see, eg., http://www.mcs.sdsmt.edu/tkowalsk/notes/Common-Taylor-series.pdf [Broken] .

RGV

Last edited by a moderator: May 5, 2017
3. Oct 3, 2011

Eight

Yeah it's come up in my 2nd year Quantum module. Thanks for the link, I think I'd just forgotten that, curse of the double gap year.

Makes sense now, thanks again.