Poisson Derivation 2

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This derivation of the Poisson Distribution Pλ(n,T) is based on generating functions, but we really only need to know about Taylor series to make this work. This one is elegant but requires a little calculus. By a little logic, we can write

P_\lambda(n,T+dT) = P_\lambda(n,T)(1-\lambda dT) + P_\lambda(n-1,T) \lambda\, dT .\,

This is a logical statement, which states that the probability of n events in T + dT is just the sum of (the probability that n already occurred up to time T) times (the probability that n event occurs in the next dT) and (the probability that n − 1 events occurred up to time T) times (the probability that the nth event occurs in the next dT). This whole derivation relies only on that logical statement, which is why it's elegant. Since dT is infinitesimal, we may take the limit dT\rightarrow 0 to find a partial differential equation
\frac{\partial }{\partial T} P_\lambda(n,T) = -\lambda\left[ P_\lambda(n,T) - P_\lambda(n-1,T)\right] .
Now let's write a generating function G_\lambda(s,T)\,
G_\lambda(s,T) = \sum_{n=0}^\infty s^n P_\lambda(n,T) .
Since we know how P_\lambda(n,T)\, evolves with T, let's try the same trick on G.
\frac{\partial}{\partial T} G_\lambda(s,T) = \sum_{n=0}^\infty s^n \frac{\partial}{\partial T} P_\lambda(n,T)

= -\lambda\left( \sum_{n=0}^\infty s^n P_\lambda(n,T) - \sum_{n=0}^\infty s^n P_\lambda(n-1,T)\right)
= -\lambda G_\lambda(s,T)+\lambda s \sum_{n=0}^\infty s^{n-1}P_\lambda(n-1,T)

=\lambda (s-1)G_\lambda(s,T)\,
where in the middle there we used the fact that Pλ(n − 1,T) is zero since we can't observe n − 1 events in any interval. In the last line, we found a simple differential equation that we can solve easily to find
G_\lambda(s,T) = G(s,0) e^{\lambda T(s-1)}.\,
You should now convince yourself that G(s,0) = 1 (how many events can occur in 0 time?). We can expand e^{\lambda T s}\, to find
G_\lambda(s,T) = e^{\lambda T(s-1)} = e^{-\lambda T} \sum_{n=0}^\infty \frac{\left(\lambda T\right)^n s^n}{n!}.
The coefficient of sn is Pλ(n,T) by definition (if you're worried about uniqueness, think about Taylor's theorem), so we have
P_\lambda(n,T) = \frac{e^{-\lambda T}(\lambda T)^n}{n!}.

References

Nicholas G. van Kampen, Stochastic processes in physics and chemistry, Elsevier Science Pub. Co., Amsterdam.
Crispin W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences, 2nd ed., Springer-Verlag.

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