Poisson Arrival Process & Exponential Inter-Arrival Waiting Time Calculator

Compute Poisson count probability P(X=k) and Exponential waiting time CDF/survival.

Poisson Probability P(X = 5)
16.062 %
Expected Events in Window (μ = λt) 6 events
Probability of ≥1 Event (1 − e^−λt) 99.752 %
Mean Inter-Arrival Wait (1 / λ) 15 minutes
Inputs
⚡ Scenarios:
events/hr
Average event rate λ per hour (or unit time).
hours
Observation window duration t in hours.
k events
Exact event count k to evaluate P(X = k).
Results & Graph
Poisson Probability P(X = 5)
16.062 %
Expected Events in Window (μ = λt)
6 events
Probability of ≥1 Event (1 − e^−λt)
99.752 %
Mean Inter-Arrival Wait (1 / λ)
15 minutes

Poisson Probability P(X = 5)

Graph 1 • Poisson & Exponential Waiting • Domain Architecture & Live Telemetry Maps Arrival Rate Lambda Per Hour, Time Window Hours T, Target Event Count K through Poisson & Exponential Waiting Formula.
Poisson Probability P(X = 5): 16.062 % Poisson & Exponential Waiting • Invariants & Metrics Poisson Probability P(X = 5) 16.062 % Expected Events in Window (μ 6 events Probability of ≥1 Event (1 − 99.752 % Mean Inter-Arrival Wait (1 / 15 minutes
Graph 2 • Poisson Probability P(X = 5) Across Arrival Rate Lambda Per Hour Sweep (40% → 160%) Evaluates P(X = k) = (λt)^k × e^(−λt) / k!; P(Wait > t) = e^(−λt); MedianWait across 7 operating points (current input highlighted).
0 % 9.61 % 19.22 % 6.02 1.60 13.77 2.40 17.48 3.20 16.06 4 12.04 4.80 7.84 5.60 4.60 6.40
Graph 3 • Joint Sensitivity: Arrival Rate Lambda Per Hour × Time Window Hours T Compares Poisson Probability P(X = 5) when Time Window Hours T is shifted +25% (solid) vs. −25% (dashed).
0 % 9.62 % 19.24 % --- Time Window Hours T @ 75% 17.44 -30% 14.95 -15% 10.94 +0% 7.14 +15% 4.28 +30%
Summary: With mean μ = λt = 6 events in 1.5 hrs, P(X = 5) is 16.06%, P(X ≥ 1) is 99.75%, and mean exponential wait between events is 15 minutes.
Summary: With mean μ = λt = 6 events in 1.5 hrs, P(X = 5) is 16.06%, P(X ≥ 1) is 99.75%, and mean exponential wait between events is 15 minutes.
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Formula & Mathematical Methodology: Poisson & Exponential Waiting Formula

P(X = k) = (λt)k × e−λt / k!
P(Wait > t) = e−λt
MedianWait = ln(2) / λ
P(X = k) = (λt)^k × e^(−λt) / k!; P(Wait > t) = e^(−λt); MedianWait = ln(2) / λ
  • arrivalRateLambdaPerHour: Arrival Rate Lambda Per Hour (events/hr)
  • timeWindowHoursT: Time Window Hours T (hours)
  • targetEventCountK: Target Event Count K (k events)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Poisson & Exponential Waiting

Compute Poisson count probability P(X=k) and Exponential waiting time CDF/survival.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P(X = k) = (λt)^k × e^(−λt) / k!; P(Wait > t) = e^(−λt); MedianWait = ln(2) / λ.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Poisson & Exponential Waiting via P(X = k) = (λt)^k × e^(−λt) / k!; P(Wait > t) = e^(−λt); MedianWait = ln(2) / λ.

Frequently Asked Questions

How does the Poisson Arrival Process & Exponential Inter-Arrival Waiting Time Calculator compute the solution?

It evaluates P(X = k) = (λt)^k × e^(−λt) / k!; P(Wait > t) = e^(−λt); MedianWait = ln(2) / λ directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Poisson Arrival Process & Exponential Inter-Arrival Waiting Time Calculator: inputs, method, and how to read the result

Compute Poisson count probability P(X=k) and Exponential waiting time CDF/survival.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Arrival Rate Lambda Per Hour: Average event rate λ per hour (or unit time).
  • Time Window Hours T: Observation window duration t in hours.
  • Target Event Count K: Exact event count k to evaluate P(X = k).

What the calculation includes

With the default values shown, Poisson Probability P(X = 5) is 16.062 %. The calculation follows Poisson & Exponential Waiting Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Arrival Rate Lambda Per Hour from 4 events/hr to 4.4 events/hr changes Poisson Probability P(X = 5) from 16.062 % to 14.197 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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