Poisson Distribution Calculator

How to Calculate Poisson Distribution Probabilities

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 average rate and independently of the time since the last event.

Poisson Distribution Formula

The probability mass function of a Poisson distribution is given by:

Where:

• \(P(X = k)\) is the probability of exactly \(k\) occurrences
• \(e\) is Euler's number (approximately 2.71828)
• \(\lambda\) (lambda) is the average number of events per interval
• \(k\) is the number of occurrences for which we want to calculate the probability
• \(k!\) is the factorial of \(k\)

Calculation Steps

1. Determine the average rate of events (\(\lambda\)) and the number of occurrences (\(k\)) you're interested in.
2. Calculate \(e^{-\lambda}\) using the exponential function.
3. Calculate \(\lambda^k\) by raising \(\lambda\) to the power of \(k\).
4. Calculate \(k!\) (the factorial of \(k\)).
5. Plug these values into the formula and compute the final probability.

Example

Let's calculate the probability of exactly 3 cars arriving at a drive-through window in a 10-minute period, given that cars arrive at an average rate of 2 per 10 minutes.

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

Therefore, the probability of exactly 3 cars arriving in a 10-minute period is about 0.1804 or 18.04%.

Visual Representation

This diagram illustrates the Poisson distribution for \(\lambda = 2\). The green bar represents the probability of exactly 3 occurrences, which is the example we calculated.