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Enter the average rate of occurrence (λ), Poisson random variable (x), and select the type of probability (exact, cumulative, or complement) to find the probability of an event happening.

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This Poisson distribution calculator finds the probability of how often an event will likely occur within a fixed interval of time or space, given an average rate of occurrences(λ). It provides a step-by-step calculation and graph for a better understanding of discrete probability distributions.

This distribution helps to predict the probability of how many times a specific number of events can occur within a fixed interval (space or time).

**Example:** Imagine counting the number of people passing through a walkthrough gate in one minute. Poisson distribution helps determine the probability of a specific number of people passing through during the defined duration.

- All the events occur independently of each other
- Two events can not occur at the same time
- Mean E(X) = Variance V(X) = λ
- The average rate of occurrence (λ) remains constant over time, where np = λ
- The value of the standard deviation is the same as the result of the square root of the mean

P(X = x) = e^{-λ}λ^{x} x!

Where:

- P(X = x) is the Probability of x occurrences
- e indicates Euler's constant (approximately 2.71828)
- λ (lambda) is the the average rate of occurrences
- x shows the number of occurrences (poisson random variable)
- x! is the factorial of x

- Determine the average rate of occurrences
- Write down the desired number of occurrences (x)
- Calculate the factorial of x
- Put values in the Poisson distribution formula, solve the exponent part
- After that divide the result by the factorial of x

Suppose you work in a call center, where you receive an average of 4 calls per minute. Calculate the following probabilities:

- P(X = 3): Probability of receiving exactly 2 calls in a minute
- P(X < 3): Probability of receiving less than 2 calls in a minute
- P(X ≤ 3): Probability of receiving at most 2 calls in a minute

**Solution:**

**Given that:**

- λ = 4 calls/minute

**Probability P(x = 3):**

Using the Poisson formula:

P(X = 3) = e^{-4}*(4)^{3} 3!

P(X = 3) = 0.018315 * 64 3 * 2 * 1

Poisson Distribution ≈ 0.19536

This means that the probability of having 3 calls is approximately 19.536 %

**Calculating the probability P(x < 3) (For less than):**

P(X = 0) = e^{-4}*(4)^{0} 0!

P(X = 0) ≈ 0.018315

P(X = 1) = e^{-4}*(4)^{1} 1!

P(X = 0) ≈ 0.07326

P(X = 2) = e^{-4}*(4)^{2} 2!

P(X = 2) ≈ 0.14652

P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.018315 + 0.07326 + 0.14652 = 0.238095

The probability of having less than 3 calls per minute is approximately 0.238095 or 23.8095%. It indicates a low probability of having less than 3 calls per minute.

**Calculate probability P(x ≤ 3) for each value of X:**

P(X = 0) ≈ 0.018315

P(X = 1) ≈ 0.07326

P(X = 2) ≈ 0.14652

P(X = 3) = e^{-4}*(4)^{3} 3!

P(X = 3) = 0.19536

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

P(X ≤ 3) ≈ 0.018315 + 0.07326 + 0.14652 + 0.19536 ≈ 0.433455

The probability of receiving less than or equal to 3 calls per minute is P(X≤ 3) ≈ 0.433455

Calculating Poisson probabilities manually can be time-consuming. To save time and simplify the calculation use our poisson distribution calculator. No matter, whether you are a beginner, student, researcher, or professional, the calculator can handle all your Poisson probability needs.

λ |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|

X | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |

0 |
0.9048 | 0.8187 | 0.7408 | 0.6703 | 0.6065 | 0.5488 | 0.4966 | 0.4493 | 0.4066 | 0.3679 |

1 |
0.0905 | 0.1637 | 0.2222 | 0.2681 | 0.3033 | 0.3293 | 0.3476 | 0.3595 | 0.3659 | 0.3679 |

2 |
0.0045 | 0.0164 | 0.0333 | 0.0536 | 0.0758 | 0.0988 | 0.1217 | 0.1438 | 0.1647 | 0.1839 |

3 |
0.0002 | 0.0011 | 0.0033 | 0.0072 | 0.0126 | 0.0198 | 0.0284 | 0.0383 | 0.0494 | 0.0613 |

4 |
0.0000 | 0.0001 | 0.0003 | 0.0007 | 0.0016 | 0.0030 | 0.0050 | 0.0077 | 0.0111 | 0.0153 |

5 |
0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0002 | 0.0004 | 0.0007 | 0.0012 | 0.0020 | 0.0031 |

6 |
0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0002 | 0.0003 | 0.0005 |

7 |
0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 |

**Poisson Distribution:**

- The variance is equal to the mean
- The number of event occurrences is counted over a fixed time interval or space
- This is suitable for events happening independently at a constant rate

**Binomial Distribution:**

- Each trial has Two possible outcomes (success or failure)
- The number of times an experiment is repeated is known

Poisson distribution is good for modeling independent events at a constant average rate within a specified interval.

Here are some general use cases:

- Counting occurrences
- Rare events
- Quality control
- Queueing systems

Easily calculate Poisson probabilities for these scenarios with the help of our Poisson distribution calculator. It can handle a variety of use cases, providing reliable results.

**Reference:**

From the source of Wikipedia: Probability mass function, Assumptions and validity.

From the source of Investopedia: Understanding Poisson Distributions.

From the source of Brilliant ORG: Conditions for Poisson Distribution, Probabilities, Properties.

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