A **Poisson distribution calculator** is a **powerful tool** used to compute **probabilities** for events that occur **randomly** over a **fixed interval** of time or space, given a known **average rate**.

Calculating the **probability** of receiving **3 customer complaints** in a day, given an average of **2 complaints** per day.

## Poisson Distribution Calculator

λ (Average) | k (Events) | P(X = k) |
---|---|---|

0.5 | 0 | 0.6065 |

0.5 | 1 | 0.3033 |

0.5 | 2 | 0.0758 |

1 | 0 | 0.3679 |

1 | 1 | 0.3679 |

1 | 2 | 0.1839 |

1.5 | 0 | 0.2231 |

1.5 | 1 | 0.3347 |

1.5 | 2 | 0.2510 |

2 | 0 | 0.1353 |

2 | 1 | 0.2707 |

2 | 2 | 0.2707 |

3 | 0 | 0.0498 |

3 | 1 | 0.1494 |

3 | 2 | 0.2240 |

3 | 3 | 0.2240 |

4 | 0 | 0.0183 |

4 | 1 | 0.0733 |

4 | 2 | 0.1465 |

4 | 3 | 0.1954 |

4 | 4 | 0.1954 |

5 | 0 | 0.0067 |

5 | 1 | 0.0337 |

5 | 2 | 0.0842 |

5 | 3 | 0.1404 |

5 | 4 | 0.1755 |

5 | 5 | 0.1755 |

6 | 0 | 0.0025 |

6 | 1 | 0.0150 |

6 | 2 | 0.0450 |

6 | 3 | 0.0900 |

6 | 4 | 0.1350 |

6 | 5 | 0.1620 |

6 | 6 | 0.1620 |

7 | 0 | 0.0009 |

7 | 1 | 0.0063 |

7 | 2 | 0.0220 |

7 | 3 | 0.0515 |

7 | 4 | 0.0903 |

7 | 5 | 0.1264 |

7 | 6 | 0.1475 |

7 | 7 | 0.1475 |

## Poisson Distribution Formula

The **Poisson distribution formula** is:

**P(X = k) = (e^-λ × λ^k) / k!**

Where:

P(X = k)is theprobabilityofk eventsoccurringeisEuler’s number(≈ 2.71828)λ(lambda) is theaverage numberof events per intervalkis thenumber of eventswe’re calculating theprobabilityfor

## How Do You Calculate the Poisson Distribution?

Determineλ (average rate of events)

Choosek (number of events you’re interested in)

Applythe formula

Simplifyandcalculatethe result

Calculating **P(X = 3)** with **λ = 2**

- λ = 2
- k = 3
- P(X = 3) = (e^-2 × 2^3) / 3!
- P(X = 3) ≈
**0.180**or**18.0%**

Calculate the probability of observing exactly **5 events**, given an average rate of **3 events** per interval, using the formula:

P(X = 5) = (e^(-3) * 3^5) / 5! = 0.1236

The probability of observing exactly **5 events**, given an average rate of **3 events** per interval, is approximately** 0.1236 or 12.36%**.

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