Poisson Distribution Calculator

Subtract the Probability from

Poisson random variable (λ):

Rate of success (x):

step:

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Poisson Distribution Calculator

This Poisson distribution calculator finds the probability of how often an event occurs within a fixed interval of time or space, given an average rate of occurrence (λ). It provides step-by-step calculations and graphs for better understanding of discrete probability distributions.

What Is Poisson Distribution?

The Poisson distribution predicts the probability of a specific number of events happening within a fixed interval of time or space.

Example: Counting the number of people passing through a gate in one minute. Poisson distribution can determine the probability of exactly a certain number of people passing through during that minute.

Properties of the Poisson Distribution:

Events occur independently of each other.
No two events occur at exactly the same time.
Mean E(X) = Variance V(X) = λ.
Average rate of occurrence (λ) is constant over time: np = λ.
Standard deviation: σ = √λ.

Poisson Distribution Formula:

P(X = x) = e^-λ λ^x x!

Where:

P(X = x) = Probability of exactly x occurrences
e = Euler’s constant (~2.71828)
λ = Average rate of occurrences in the interval
x = Number of occurrences (Poisson random variable)
x! = Factorial of x

How to Calculate Poisson Distribution:

Determine the average rate of occurrences (λ).
Decide the desired number of occurrences (x).
Calculate the factorial of x (x!).
Substitute values into the Poisson formula and evaluate e^-λ λ^x.
Divide the result by x! to get P(X = x).

Example: Call Center

Average calls received: 4 per minute (λ = 4). Find:

P(X = 3): Probability of exactly 3 calls per minute
P(X < 3): Probability of fewer than 3 calls
P(X ≤ 3): Probability of at most 3 calls

1. Probability P(X = 3):

P(X = 3) = (e^-4 * 4³) / 3! = (0.018315 * 64) / 6 ≈ 0.19536

≈ 19.54%

2. Probability P(X < 3):

Sum probabilities for X = 0, 1, 2:

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

P(X < 3) ≈ 0.2381 (≈ 23.81%)

3. Probability P(X ≤ 3):

Include X = 3:

P(X = 0) ≈ 0.018315
P(X = 1) ≈ 0.07326
P(X = 2) ≈ 0.14652
P(X = 3) ≈ 0.19536

P(X ≤ 3) ≈ 0.43345 (≈ 43.35%)

Poisson Distribution Table (Sample)

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