How to derive the Poisson p.m.f.

[lamsung]

Can anyone derive the p.m.f. of Poisson distribution without mentioning the binomial distribution?

The binomial deriving method put lambda = np and finally the binomial p.m.f. become the Poisson one as n goes to infinity.
It seems that this is only proving that binomial distribution will approach the Poisson distribution as n goes to infinity, p goes to 0, and lambda stays constant, but it has nothing to do with deriving the p.m.f. of Poisson distribution.
So, the method has not solved my question that how does the p.m.f. of Poisson distribution come from.

I am doubtful for this.

 

[chisigma]

Can anyone derive the p.m.f. of Poisson distribution without mentioning the binomial distribution?

The binomial deriving method put lambda = np and finally the binomial p.m.f. become the Poisson one as n goes to infinity.
It seems that this is only proving that binomial distribution will approach the Poisson distribution as n goes to infinity, p goes to 0, and lambda stays constant, but it has nothing to do with deriving the p.m.f. of Poisson distribution.
So, the method has not solved my question that how does the p.m.f. of Poisson distribution come from.

I am doubtful for this.

I strongly suspect that Your consideration are derived from... Poisson Distribution -- from Wolfram MathWorld

Now 'Monster Wolfram' is a monster but that doesn't mean that all it writes is good!... the binomial distribution extablishes that the probability to have k 'good' results in n trials is...

$\displaystyle P_{n,k} = \binom {n}{k} p^{k}\ (1-p)^{n-k}\ (1)$

... and the Poisson distribution extablishes that if $\displaystyle \lambda$ is the mean number that a 'good result' occours in a unit time, then the probability to have k 'good results' in a unit time is... $\displaystyle P_{\lambda,k} = \frac{\lambda^{k}}{k!}\ e^{- \lambda}\ (2)$

In my opinion the best is to consider thye binomial and Poisson distribution as two different way to describe the reality and no more...

Kind regards

$\chi$ $\sigma$

 

[Jameson]

I am not totally sure about this (still fairly sure) but I believe that historically the Poisson Distribution was derived from the Binomial Distribution. As you said it is an extreme case of a situation with a very low $p$ and a very high $n$, so using the formulas for the binomial distribution stops becoming the most efficient or best way to describe the distribution.

You seem to already know the algebra behind the derivation, but here it is in case anyone is interested. It's a bit long to type out here from scratch.

 

[Evgeny.Makarov]

What about this document (PDF)? At least it does not mention binomial distribution.

Also, aren't there three axioms of Poisson processes from which the distribution can be derived?

 

[lamsung]

I am not totally sure about this (still fairly sure) but I believe that historically the Poisson Distribution was derived from the Binomial Distribution. As you said it is an extreme case of a situation with a very low $p$ and a very high $n$, so using the formulas for the binomial distribution stops becoming the most efficient or best way to describe the distribution.

You seem to already know the algebra behind the derivation, but here it is in case anyone is interested. It's a bit long to type out here from scratch.

So, can I say the following?

1. Poisson distribution is actually a (extreme case of) Binomial distribution.

2. If X ~ Po(lambda), then X ~ B(n, lambda/n) for a large n.

3. Using Poisson distribution but not Binomial distribution is due to efficiency of calculation. In other words, Poisson distribution is used to estimate the Binomial distribution provided the expectation (that is, np).

4. Suppose "success" is randomly distributed in a time interval (say, 3 unit time), disjoint regions are independent. Then lambda = number of success / 3. We can then use the Poisson distribution to find out the probability of number of success in a unit time, no matter how big (or how small) lambda is.

 

[lamsung]

What about this document (PDF)? At least it does not mention binomial distribution.

Also, aren't there three axioms of Poisson processes from which the distribution can be derived?

Thanks for sharing. I used a day to understand the document.
It almost solves my question, except, I am quite unsure about the first equation.
The equation states that the probability of one event occurs in a short interval (delta t) equals to lambda times delta t.
Intuitively, I agree. But I want a proof.

 

[Evgeny.Makarov]

The equation states that the probability of one event occurs in a short interval (delta t) equals to lambda times delta t.
Intuitively, I agree. But I want a proof.

I believe this is an assumption from which you derive the Poisson distribution. There are other distributions for which it does not hold.