What is Binomial and Poisson process?

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Binomial Process:

Binomial process is the number of successes in the first n independent Bernoulli trials, where n = 0,1,2, . . .

It is a discrete-time discrete-space counting stochastic process.


λ = arrival rate

Δ = frame size

P = Probability of success during one frame trial

x(t/Δ) = Number of arrivals by the time t

T = inter arrival time

The inter arrival period consists of a Geometric number of frames Y, each frame taking Δ seconds. Hence the inter arrival time can be computed as T = YΔ. It is rescaled Geometric random variable taking possible values, Δ, 2Δ, 3Δ . …

λ = p/Δ

n = t/Δ

X(n) = Binomial (n, p)

Y = Geometric (p)

T = YΔ

E(t) = E(YΔ) = ΔE(Y) = Δ/p = 1/λ

V(T) = V(YΔ) = Δ2V(Y) = \((1-p)\left ( \frac{Δ}{p} \right )^2 = \frac{1-p}{λ^2}\)

Poisson Process:

It is limiting case of Binomial process as binomial counting process Δ tends to 0. Poisson process is a continuous time counting stochastic process obtained from Binomial counting process when its frame size Δ decreases to 0 while the arrival rate λ remains constant.

Let X(t) = No. of arrivals occurring until time t

T = inter arrival time

Tk = time of Kth arrival

X(t) = Exponential(λ)

Tk = Gamma (k, λ)

E X(t) = np = \(\frac{t}{Δ}p\) = λt

V X(t) = λt

Fr(t) = 1-eλt

Probability of Kth arrival before time t

P(Tk ≤ t) = P[X(t) ≥ k]

P(Tk > t) = P[X(t) < k]

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