Poisson process rates and event probabilities
A homogeneous Poisson process models random events occurring over time at a constant average rate \(\lambda\)
(events per unit time). Typical applications include arrivals (customers/calls), breakdowns, and random counts.
1) Count of events in time \(t\)
Let \(N(t)\) be the number of events observed in a time window of length \(t\).
For a Poisson process with rate \(\lambda\),
\[
N(t)\sim \mathrm{Poisson}(\mu),\qquad \mu=\lambda t.
\]
The probability of observing exactly \(k\) events is:
\[
P(N(t)=k)=e^{-\mu}\frac{\mu^k}{k!}
=e^{-\lambda t}\frac{(\lambda t)^k}{k!},\qquad k=0,1,2,\dots
\]
Mean and variance
\[
E[N(t)] = \mu=\lambda t,\qquad \mathrm{Var}(N(t))=\mu=\lambda t.
\]
2) Cumulative probabilities: “at most” and “at least”
Because \(N(t)\) is a discrete count (it can only be \(0,1,2,\dots\)),
cumulative probabilities are computed by summing the PMF.
At most \(k\) events
\[
P(N(t)\le k)=\sum_{i=0}^{k} e^{-\mu}\frac{\mu^i}{i!}.
\]
In a discrete distribution, \(P(N(t)
At least \(k\) events
The cleanest way is to use the complement:
“at least \(k\)” means “not below \(k\)”.
\[
P(N(t)\ge k)=1-P(N(t)\le k-1)
=1-\sum_{i=0}^{k-1} e^{-\mu}\frac{\mu^i}{i!}.
\]
Example: \(P(N(t)\ge 3)=1-[P(N(t)=0)+P(N(t)=1)+P(N(t)=2)]\).
3) Waiting times and the exponential distribution
Let \(T_1\) be the time until the first event. In a Poisson process,
\[
T_1 \sim \mathrm{Exp}(\lambda),
\qquad
P(T_1\le t)=1-e^{-\lambda t},
\qquad
P(T_1>t)=e^{-\lambda t}.
\]
Notice that \(P(T_1>t)\) is exactly the probability of zero arrivals by time \(t\):
\(P(N(t)=0)=e^{-\lambda t}\).
Memoryless property
\[
P(T_1>s+t\mid T_1>s)=P(T_1>t).
\]
4) Simulation perspective (timeline)
A practical way to simulate a Poisson process is to generate exponential interarrival times
\(S_1,S_2,\dots\sim \mathrm{Exp}(\lambda)\) and form event times:
\[
\tau_1=S_1,\quad \tau_2=S_1+S_2,\quad \dots
\]
The calculator’s “Play” button animates a sample path, showing event marks along the time axis.
5) Extensions (university level)
In a non-homogeneous Poisson process, the rate depends on time: \(\lambda=\lambda(t)\). Then:
\[
\mu(t)=\int_0^t \lambda(u)\,du,
\qquad
N(t)\sim \mathrm{Poisson}(\mu(t)).
\]
6) What this calculator computes
- Count probabilities \(P(N(t)=k)\), \(P(N(t)\le k)\), and \(P(N(t)\ge k)\) using \(\mu=\lambda t\).
- Waiting time probabilities using \(T_1\sim \mathrm{Exp}(\lambda)\).
- An optional simulation preview to estimate probabilities and animate an arrival timeline.
