The Poisson distribution models rare events in fixed intervals, like accidents per day or defects per product batch. It's based on the law of rare events, assuming independent occurrences at a constant average rate. This distribution is crucial for analyzing count data in various fields.

Calculating Poisson probabilities involves the probability mass function, which depends on the average event rate and number of events. The Poisson distribution can also approximate the binomial distribution under specific conditions, simplifying calculations for large sample sizes with rare events.

Poisson Distribution

Poisson distribution for fixed intervals

• Models the number of events occurring in a fixed interval of time (hour, day) or space (square meter, cubic kilometer)
• Events are independent and occur at a constant average rate ($\lambda$) within the interval
• Rate parameter $\lambda$ represents the average number of events in the fixed interval
• Probability of a specific number of events depends on $\lambda$ and the number of events ($x$)
• Appropriate when events are rare relative to interval size, occur independently, and have a constant average rate
• Based on the law of rare events, which states that the total number of events follows a Poisson distribution if events occur independently and at a constant rate

Probability calculations with Poisson

• Probability mass function (PMF) for Poisson distribution: $P(X = x) = \frac{e^{-\lambda}\lambda^x}{x!}$

  • $X$ is the random variable for number of events
  • $x$ is the specific number of events (non-negative integer)
  • $e$ is the mathematical constant (2.71828)
  • $\lambda$ is the average number of events in the fixed interval

• Calculate probability by substituting $x$ and $\lambda$ into PMF formula

• Cumulative distribution function (CDF) calculates probability of observing at most a certain number of events

  • CDF is the sum of PMF values from 0 to specified $x$ value

• Statistical software and calculators have built-in functions for Poisson probabilities

Poisson vs binomial approximation

• Poisson distribution can approximate binomial distribution under certain conditions
• Binomial distribution models number of successes in fixed number of independent trials with same success probability
• Poisson approximation is appropriate when:
  1. Number of trials ($n$) is large (>20)
  2. Probability of success ($p$) in each trial is small (<0.1)
  3. Product of $n$ and $p$ is constant, denoted as $\lambda$ ($n \times p = \lambda$)
• Under these conditions, binomial distribution converges to Poisson distribution
  • Poisson parameter $\lambda$ equals product of binomial parameters $n$ and $p$
• Approximating binomial with Poisson simplifies calculations when conditions are met
  • Useful when number of trials is very large, making binomial calculations more complex

Properties of Poisson Distribution

• Mean and variance of a Poisson distribution are both equal to $\lambda$
• Poisson is a discrete probability distribution, as it models count data
• Related to the exponential distribution, which models the time between Poisson events
• Exhibits the memoryless property, meaning the probability of future events is independent of past events