Poisson Probability Calculator
Calculate the probability of observing exactly k events in a fixed interval when events occur at a known average rate lambda.
P(X = k)
0.14652511
P(X = k) vs Average Rate (lambda)
Formula
## How to Calculate Poisson Probability ### Formula **P(X = k) = (lambda^k * e^(-lambda)) / k!** The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate lambda. Both the mean and variance equal lambda.
Esempio Risolto
A call center receives an average of 4 calls per hour. What is the probability of exactly 2 calls in an hour?
- 01lambda = 4, k = 2
- 02P(X=2) = (4^2 * e^(-4)) / 2!
- 03= (16 * 0.01832) / 2
- 04= 0.29305 / 2
- 05= 0.14653
Domande Frequenti
When should I use the Poisson distribution?
Use it when counting events in a fixed interval (time, area, volume), events are independent, the rate is constant, and two events cannot occur at exactly the same instant. Examples: emails per hour, defects per meter, accidents per year.
How does Poisson relate to the binomial?
The Poisson distribution is the limit of the binomial as n approaches infinity and p approaches zero while np = lambda stays constant. It approximates the binomial well when n >= 20 and p <= 0.05.
Can lambda be a decimal?
Yes. Lambda represents the average rate and can be any positive real number. For example, 2.5 emails per minute is perfectly valid.
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