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2.5.1 Poisson Random Variable and its Distribution

Introduction

  • The Poisson random variable is a discrete variable used to model the number of events occurring within a fixed interval of time or space.
  • It is named after the French mathematician Siméon Denis Poisson.

Characteristics

  1. Discreteness:
  2. Like Bernoulli and binomial distributions, the Poisson is a discrete distribution but usually over an infinite range (0, 1, 2, ...).
  3. Single Parameter (λ):
  4. Unlike the binomial distribution, which is described by two parameters (n and p), the Poisson distribution is characterized by a single parameter λ, which represents the average rate of events per interval.
  5. Probability Mass Function (PMF): image

where: - k is the number of occurrences, - e is the base of the natural logarithm, - λ is the average number of events in the given interval.

Key Properties

  1. Mean and Variance:
  2. For a Poisson distribution, the mean and variance are both equal to \( \lambda \).
  3. Thus, the standard deviation is image .
  4. No Memory:
  5. The Poisson process has the property that the probability of a given number of events occurring in any interval is independent of any occurrences outside this interval.

Examples

  1. Customer Arrivals: Counting the number of customers arriving at a store in a specific time period.
  2. Call Volume: Number of calls received at a call center within a certain timeframe.
  3. Manufacturing: Number of defects found in a specified length of material.

Practical Uses

  1. Event Occurrences:
  2. Well-suited for modeling random events that happen independently of each other within a given period.
  3. Various Fields:
  4. Used across various domains like:
    • Telecommunication (for call arrivals),
    • Astronomy (for photon arrivals from a light source),
    • Traffic engineering (for vehicle arrivals at a stop).

Additional Insights

  • The Poisson distribution can often serve as an approximation to the binomial distribution when:
  • n (number of trials) is large,
  • p (probability of success) is small,
  • np (mean) is moderate.
  • It simplifies dealing with events occurring over time or space without having to track the exact intervals between events, unlike more complex models that require detailed interval data.
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