Poisson Distribution

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

Properties

Mean and Variance

• Mean: λ
• Variance: λ

Mode, Skewness, and Kurtosis

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

Applications

• Modeling defects in manufacturing processes
• Analyzing accidents or errors in a given period
• Modeling customer arrivals at service facilities
• Analyzing mutations in DNA sequences

Relationship with Other Distributions

• Limiting case of the binomial distribution
• Special case of the negative binomial distribution
• Related to the exponential and gamma distributions