Cumulative Poisson Distribution Calculator

What is the Cumulative Poisson Distribution?

The Cumulative Poisson Distribution calculates the probability that the number of events occurring in a fixed interval of time or space is less than or equal to a specified value, given a known average rate of occurrence.

Formula and Its Meaning

The formula for the Cumulative Poisson Distribution is:

\[P(X \leq x) = \sum_{k=0}^{x} \frac{e^{-\lambda} \lambda^k}{k!}\]

Where:

• \(P(X \leq x)\) is the cumulative probability
• \(e\) is Euler's number (approximately 2.71828)
• \(\lambda\) (lambda) is the average rate of events in the interval
• \(x\) is the maximum number of events we're interested in
• \(k\) is the variable that takes values from 0 to x

Calculation Steps

1. Determine the values of λ and x.
2. Calculate the probability for each value of k from 0 to x using the Poisson probability mass function.
3. Sum these probabilities to get the cumulative probability.

Example Calculation

Let's calculate the cumulative Poisson probability for λ = 2 and x = 3:

1. P(X = 0): \(\frac{e^{-2} 2^0}{0!} = 0.1353\)
2. P(X = 1): \(\frac{e^{-2} 2^1}{1!} = 0.2707\)
3. P(X = 2): \(\frac{e^{-2} 2^2}{2!} = 0.2707\)
4. P(X = 3): \(\frac{e^{-2} 2^3}{3!} = 0.1804\)
5. P(X ≤ 3) = 0.1353 + 0.2707 + 0.2707 + 0.1804 = 0.8571

Visual Representation

This bar chart represents the Poisson probabilities for x = 0 to 3 when λ = 2. The cumulative probability P(X ≤ 3) is the sum of these individual probabilities.