Poisson Distribution Calculator
Calculate exact and cumulative Poisson distribution probabilities instantly. Enter your known average rate (Lambda) and target occurrences to generate a complete probability breakout matrix.
The average number of expected occurrences (lambda).
The exact number of successes you want the probability for.
Mastering the Poisson Equation
The Poisson distribution is one of the most powerful tools in statistical analysis. It allows data scientists to estimate the probability of events occurring in the future based strictly on the average of what happened in the past. It assumes that each event is completely independent (e.g. the arrival of one customer at a coffee shop does not physically cause the arrival of the next customer).
The Fundamental Probability Mass Function (PMF)
Where:
- P(X = x): The exact probability that the event occurs exactly
xtimes. - λ (Lambda): The average number of events that historically happen in that timeframe.
- e: Euler's number, a fundamental mathematical constant (~2.71828).
- x!: The factorial of x (e.g., 3! = 3 × 2 × 1).
Example Application (Hospital ER)
Imagine a hospital emergency room that historically receives an average of 4 patients per hour during the midnight shift. The hospital administrator wants to know: What is the probability that exactly 6 patients will arrive between midnight and 1:00 AM tonight?
- 1. Define Variables: Lambda (λ) = 4, x = 6.
- 2. Substitute into formula: P(X = 6) = (46 × e-4) / 6!
- 3. Calculate Numerator: 4096 × 0.018315 = 75.021
- 4. Calculate Denominator: 6! = 720
- 5. Divide to solve: 75.021 / 720 = 10.42%
Frequently Asked Questions
Studying Statistics?
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