Definition of Problem
To model probability distribution P(X=n, t) of noises caused by random arrival of electrons in a vacum tube.
λ = Probability constant of electron arrive over a period of time, intensity of Poisson Dist.
n = Total number of electrons arrived at time t
G = Generating function
Let P(n, t) be the probability of a total of n electrons arrived at time t. The probability of an electron arriving over a period Δt can be modulated by a constant λ:
Then the probability of having n arrival at t + Δt is the sum of probabilities: 1) Already has n arrival and no arrivals in Δt. 2) Already has n – 1 arrival and one arrivals in Δt. Formulated as follow:
Re-arrange the upper equation we get [Eq.1]
At this point, we would like to introduce a math skill Generating Function and denotes:
[P(-1, t) = 0 for all t since number of arrival must not be negative]
We then get ourselves a p.d.e:
With the initial condition of G(s, 0) being 1 because P(0, 0) = 1 and P(n, 0) = 0 for all n, the particular solution would be:
Perform Taylor expansion at s = 0:
By comparing terms in powers of s, we can deduce:
which is actually a Poisson Process.