Poisson Distribution Calculator

Calculate probabilities for rare events using the Poisson distribution. Perfect for analyzing event occurrences in fixed intervals of time or space.

Understanding Poisson Distribution

Definition: Discrete probability distribution for rare events

Mean (λ): Average rate of events in the interval

Variance: Equal to the mean in Poisson distribution

Key Properties

Independent Events: Occurrences don't affect each other

Fixed Interval: Time or space period must be constant

Rare Events: Individual probability is small

Theoretical Foundation

The Poisson distribution emerges as a limiting case of the binomial distribution when the number of trials approaches infinity while the product of trials and probability remains constant. This mathematical relationship provides insight into the distribution's fundamental nature as a model for rare events occurring in continuous time or space. The distribution's unique properties arise from its connection to the exponential distribution and point processes.

The theoretical underpinning of the Poisson distribution rests on the concept of complete spatial randomness and the memoryless property. These mathematical principles ensure that events occur independently and that the probability of an event in any interval depends only on the interval's length, not its position or history.

Mathematical Framework

The probability mass function of the Poisson distribution is defined by:

P(X = k) = (e^(-λ) × λ^k) / k!

Where:

  • λ = Mean rate of events
  • k = Number of events
  • e = Euler's number
  • k! = Factorial of k

Key Properties:

  • E(X) = λ
  • Var(X) = λ
  • Skewness = 1/√λ
  • Kurtosis = 1/λ

Moment Generating Function

The moment generating function (MGF) provides a powerful tool for analyzing the Poisson distribution's properties:

M(t) = exp(λ(e^t - 1))

Derivatives yield moments:

  • M'(0) = λ (First moment)
  • M''(0) = λ + λ² (Second moment)
  • M'''(0) = λ + 3λ² + λ³ (Third moment)

This function uniquely characterizes the distribution and facilitates the derivation of its statistical properties through differentiation.

Probability Generating Function

The probability generating function (PGF) offers an alternative representation:

G(z) = exp(λ(z - 1))

Properties:

  • G(1) = 1 (Probability sum)
  • G'(1) = λ (Expected value)
  • G''(1) = λ² + λ (Second factorial moment)

The PGF provides a convenient method for calculating probabilities and moments, particularly useful in analyzing sums of independent Poisson variables.

Computational Considerations

The numerical computation of Poisson probabilities requires careful handling of factorial calculations and exponential terms. For large values of λ or k, direct computation using the probability mass function can lead to numerical overflow or underflow. Modern implementations often use logarithmic transformations and specialized algorithms for numerical stability:

ln(P(X = k)) = -λ + k×ln(λ) - ln(k!)

P(X = k) = exp(ln(P(X = k)))

These computational methods ensure accurate probability calculations across a wide range of parameter values while maintaining numerical precision.