The Poisson distribution is a key player in modeling rare events and counting occurrences in fixed intervals. It's a handy tool for predicting things like customer arrivals, defects in materials, or even radioactive particle emissions. With just one parameter, lambda, it packs a punch in various fields.

This distribution fits snugly into the family of discrete distributions alongside its cousins, the Bernoulli and Binomial. While Bernoulli deals with single trials and Binomial with fixed numbers of trials, Poisson shines when counting events over time or space with no upper limit.

The Poisson Distribution

Definition and Probability Mass Function

• Poisson distribution models the number of events occurring in a fixed interval of time or space given a known average rate
• Probability mass function (PMF) expressed as $P(X = k) = (\lambda^k e^{-\lambda}) / k!$
  • $\lambda$ represents the average rate of occurrence
  • $k$ denotes the number of events
• Events occur independently and at a constant average rate
• PMF defined for non-negative integer values of $k$
• Models rare events (radioactive particle emissions, customer arrivals)
• Sum of independent Poisson-distributed random variables also Poisson-distributed
  • Parameter equals the sum of individual parameters

Applications and Properties

• Used in various fields (physics, biology, finance)
• Models queuing systems (customers in line, calls to a call center)
• Describes random spatial distributions (stars in the sky, defects in materials)
• Useful for modeling rare diseases or accidents
• Approximates binomial distribution under certain conditions
• Exhibits memoryless property
  • Probability of future events independent of past events

Parameters of the Poisson Distribution

Key Parameters and Characteristics

• Single parameter $\lambda$ (lambda) represents both mean and variance
• Expected value (mean) of Poisson-distributed random variable $X$ equals $E[X] = \lambda$
• Variance of Poisson-distributed random variable $X$ equals $Var(X) = \lambda$
• Standard deviation calculated as $\sigma = \sqrt{\lambda}$
• Right-skewed for small $\lambda$ values, more symmetric as $\lambda$ increases
• Mode equals largest integer less than or equal to $\lambda$
• Approximates normal distribution as $\lambda$ approaches infinity
  • Mean and variance both equal to $\lambda$

Interpreting Lambda

• $\lambda$ represents average number of events in the given interval
• Determines shape and spread of the distribution
• Larger $\lambda$ values lead to more symmetric distributions
• Smaller $\lambda$ values result in more skewed distributions
• Can be estimated from historical data or theoretical considerations
• Affects probability calculations and statistical inferences
• Crucial for accurate modeling and predictions in Poisson processes

Probabilities and Moments of the Poisson Distribution

Probability Calculations

• Calculate probabilities for specific $k$ values using PMF
  • $P(X = k) = (\lambda^k e^{-\lambda}) / k!$
• Use cumulative distribution function (CDF) for probability ranges
  • $P(X \leq k) = \sum_{i=0}^k (\lambda^i e^{-\lambda}) / i!$
• Moment generating function (MGF) expressed as $M(t) = e^{\lambda(e^t - 1)}$
• Higher-order moments derived from MGF or direct calculation
  • Second moment $E[X^2] = \lambda^2 + \lambda$
• Skewness calculated as $1/\sqrt{\lambda}$
  • Indicates less skew as $\lambda$ increases
• Excess kurtosis equals $1/\lambda$
  • Approaches normal distribution (excess kurtosis of 0) as $\lambda$ increases
• Statistical software or tables often used for complex calculations
  • Especially useful for large $\lambda$ or $k$ values

Practical Applications

• Calculate probabilities of specific numbers of events (customer arrivals, defects)
• Determine likelihood of rare occurrences (mutations, accidents)
• Estimate waiting times in queuing systems
• Analyze reliability of systems or components
• Model insurance claims or financial risks
• Predict number of calls to emergency services
• Optimize inventory management based on demand patterns

Poisson vs Binomial Distributions

Relationship and Similarities

• Poisson distribution derived as limiting case of binomial distribution
  • Occurs when $n$ approaches infinity and $p$ approaches 0
  • $np$ remains constant
• Poisson parameter $\lambda$ equivalent to $np$ in binomial distribution
• Poisson approximates binomial when $n$ large (typically $n > 20$) and $p$ small (typically $p < 0.05$)
• Good approximation when $n \geq 20$ and $np \leq 10$
• Law of Rare Events applies to both distributions
  • Total events follow Poisson distribution for large trials with small individual probabilities
• Both discrete distributions with non-negative integer values
• Model count data in different scenarios

Key Differences

• Binomial models successes in fixed number of trials
• Poisson models occurrences in fixed interval of time or space
• Binomial has fixed upper limit on number of events
• Poisson has no upper limit on number of events
• Binomial requires two parameters ($n$ and $p$)
• Poisson requires only one parameter ($\lambda$)
• Binomial variance less than or equal to mean
• Poisson variance always equal to mean
• Binomial approaches normal distribution as $n$ increases
• Poisson approaches normal distribution as $\lambda$ increases