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Poisson Distribution

Definition / Introduction

  • The Poisson distribution is a Discrete Probability Distribution that models the probability of a given number of events occurring in a fixed interval of time or space.
  • It's used when these events occur independently and with a known constant average rate (\(\lambda\)).
  • Examples: Number of emails received per hour, number of typos per page, number of cars arriving at a toll booth per minute, number of mutations in a DNA strand.

Key Concepts

1. Conditions for Poisson Distribution

A process can often be modeled by a Poisson distribution if: * Events occur independently (occurrence of one event doesn't affect the probability of another). * The average rate (\(\lambda\)) at which events occur is constant for the interval of interest. * The probability of an event occurring is proportional to the length of the interval (for small intervals). * Two events cannot occur at exactly the same instant (the probability of simultaneous events is negligible).

2. The Poisson Random Variable

  • A Random Variable \(X\) follows a Poisson distribution if it represents the count of events occurring in a fixed interval, given the average rate \(\lambda\).
  • Possible values for \(X\) are non-negative integers \(\{0, 1, 2, 3, ...\}\).
  • Notation: \(X \sim \text{Poisson}(\lambda)\) or \(X \sim \text{Poi}(\lambda)\).

3. Probability Mass Function (PMF)

  • The PMF gives the probability of observing exactly \(k\) events in the interval: $$ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, ... $$
  • Where:
    • \(k\) is the number of events (a non-negative integer).
    • \(\lambda\) (lambda) is the average number of events per interval (the rate parameter). \(\lambda > 0\).
    • \(e\) is Euler's number (the base of the natural logarithm, \(e \approx 2.71828...\)).
    • \(k!\) is the factorial of \(k\).

4. Parameter

  • The Poisson distribution has only one parameter:
    • \(\lambda\) (lambda): The average rate or mean number of events in the given interval.

5. Expected Value (Mean)

  • The expected number of events in the interval is simply the rate parameter: $$ E[X] = \lambda $$

6. Variance

  • A key property of the Poisson distribution is that its variance is equal to its mean: $$ Var(X) = \lambda $$
  • If the observed variance in count data is much larger than the mean (overdispersion), a simple Poisson model might not be appropriate (consider [[09_Negative_Binomial_Distribution|Negative Binomial]] instead).

7. Adjusting the Rate Parameter \(\lambda\)

  • If the average rate is given for one interval length, you must adjust \(\lambda\) proportionally if you are interested in a different interval length.
  • Example: If a call center receives an average of 10 calls per hour (\(\lambda=10\) for 1 hour), the average number of calls in a 3-hour period would be \(\lambda = 10 \times 3 = 30\). The average number of calls in a 15-minute (0.25 hour) period would be \(\lambda = 10 \times 0.25 = 2.5\).

Connections to Other Topics

  • Binomial Approximation: The Poisson distribution approximates the Binomial Distribution \(B(n, p)\) when \(n\) is large and \(p\) is small (rare events), such that \(\lambda \approx np\). This is useful because the Poisson PMF is often easier to compute than the Binomial PMF under these conditions.
  • Exponential Distribution: If the number of events occurring in an interval follows a Poisson distribution with rate \(\lambda\), then the time between consecutive events follows an Exponential Distribution with the same rate parameter \(\lambda\). They describe different aspects of the same underlying "Poisson process".

Summary

  • Models the number of events (\(k\)) in a fixed interval of time/space.
  • Assumes events occur independently at a constant average rate (\(\lambda\)).
  • Parameter: \(\lambda\) (average rate/mean number of events).
  • PMF: \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\).
  • Mean: \(E[X] = \lambda\).
  • Variance: \(Var(X) = \lambda\) (Mean = Variance).
  • Related to Binomial (rare events) and Exponential (time between events) distributions.

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