Poisson Distribution Calculator

Poisson Distribution is evaluated from Average Rate - mean number of events in interval and Number of Events. The calculation reports P - Exactly k events, P - At most k events and P - At least k events.

Results

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About the Poisson Distribution Calculator

Poisson Distribution is treated here as a quantitative relation between Average Rate - mean number of events in interval and Number of Events and P - Exactly k events, P - At most k events, P - At least k events and Mean.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
P(X=k) = (λ^k x e^-λ) / k!
Cumulative P(X<=k) = sum of P(X=i) for i=0 to k.

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

P(X=k) = (λ^k x e^-λ) / k!
Cumulative P(X<=k) = sum of P(X=i) for i=0 to k.

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Call center: Average 5 calls/minute. P(exactly 3 calls in 1 minute)

Inputs

lambda: 5 k: 3
P - Exactly k events: 0.140374. P - At most k events: 0.265026. P - At least k events: 0.875348. Mean: 5. Variance: 5

With Average Rate - mean number of events in interval = 5 and Number of Events = 3 as the stated inputs, the result is P - Exactly k events = 0.140374, P - At most k events = 0.265026 and P - At least k events = 0.875348. Each value corresponds to the declared output fields.

Example 2: Website traffic: Average 2 purchases per hour. P(4 purchases in 1 hour)

Inputs

lambda: 2 k: 4
P - Exactly k events: 0.090224. P - At most k events: 0.947347. P - At least k events: 0.142877. Mean: 2. Variance: 2

With Average Rate - mean number of events in interval = 2 and Number of Events = 4 as the stated inputs, the result is P - Exactly k events = 0.090224, P - At most k events = 0.947347 and P - At least k events = 0.142877. Each value corresponds to the declared output fields.

Example 3: Quality control: Average 1.5 defects per 100 units. P(0 defects in a 100-unit batch)

Inputs

lambda: 1.5 k: 0
P - Exactly k events: 0.22313. P - At most k events: 0.22313. P - At least k events: 1. Mean: 1.5. Variance: 1.5

With Average Rate - mean number of events in interval = 1.5 and Number of Events = 0 as the stated inputs, the result is P - Exactly k events = 0.22313, P - At most k events = 0.22313 and P - At least k events = 1. Each value corresponds to the declared output fields.

Example 4: Traffic accidents: Average 3 accidents per month at an intersection. P(5 or more accidents)

Inputs

lambda: 3 k: 5
P - Exactly k events: 0.100819. P - At most k events: 0.916082. P - At least k events: 0.184737. Mean: 3. Variance: 3

With Average Rate - mean number of events in interval = 3 and Number of Events = 5 as the stated inputs, the result is P - Exactly k events = 0.100819, P - At most k events = 0.916082 and P - At least k events = 0.184737. Each value corresponds to the declared output fields.

Common Use Cases

  • Calculate probability of rare events in a fixed interval
  • Model customer arrivals, website traffic, calls per hour
  • Quality control for defect rates