Poisson Distribution

Study Notes

Definition

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, where the events occur independently and at a constant average rate.

Key Characteristics

Discrete distribution
Models count data (number of events)
Events occur independently
Events occur at a constant average rate (λ)
Variance equals the mean (λ)

Formula

The probability mass function of the Poisson distribution is:

P(X = k) = (e^(-λ) * (λ^k)) / k!

where:

P(X = k) is the probability of k events occurring
e is the base of the natural logarithm
λ is the average rate of events (expected value)
k is the number of events

Properties

Mean: λ
Variance: λ
Mode: λ (when λ is an integer)
Skewness: 1 / √λ
Kurtosis: 1 / λ

Applications

Modeling the number of defects in a manufacturing process
Analyzing the number of accidents or errors in a given period
Modeling the number of customers arriving at a service facility
Analyzing the number of mutations in a DNA sequence

Relationship with Other Distributions

The Poisson distribution is a limiting case of the binomial distribution
The Poisson distribution is a special case of the negative binomial distribution
The Poisson distribution is related to the exponential distribution and the gamma distribution

Poisson Distribution

Models the number of events occurring in a fixed interval of time or space
Events occur independently and at a constant average rate (λ)

Key Characteristics

Discrete distribution that models count data (number of events)
Events occur independently and at a constant average rate (λ)
Variance equals the mean (λ)

Formula

Probability mass function: P(X = k) = (e^(-λ) * (λ^k)) / k!
k is the number of events, λ is the average rate of events (expected value), e is the base of the natural logarithm