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.

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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):

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.

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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 .
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.

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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.

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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).

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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.