# Definition of Problem

To model probability distribution P(X=n, t) of noises caused by random arrival of electrons in a vacum tube.

# Annotation

*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

# Derivation

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:

so that:

[*P(-1, t) *= 0 for all *t* since number of arrival must not be negative]

We then get ourselves a p.d.e:

separating variables,

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.