Poisson Distribution Calculator

What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event.

Formula and Its Meaning

The Poisson probability mass function is given by:

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

Where:

• \(e\) is Euler's number (approximately 2.71828)
• \(\lambda\) (lambda) is the average number of events per interval
• \(k\) is the number of events we're calculating the probability for
• \(k!\) is the factorial of k

Calculation Steps

1. Determine the value of λ (lambda) - the average rate of events.
2. Choose the value of k - the number of events you're interested in.
3. Calculate \(e^{-\lambda}\).
4. Calculate \(\lambda^k\).
5. Calculate \(k!\).
6. Apply the formula: \(P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\).

Example Calculation

Let's calculate the probability of exactly 3 events occurring if the average is 2 events per interval.

1. λ = 2, k = 3
2. \(e^{-\lambda} = e^{-2} \approx 0.1353\)
3. \(\lambda^k = 2^3 = 8\)
4. \(k! = 3! = 6\)
5. \(P(X = 3) = \frac{0.1353 \times 8}{6} \approx 0.1804\)

Visual Representation

This bar chart represents a Poisson distribution with λ = 2. The red bar highlights the probability for k = 3 events.