The Poisson distribution is a discrete probability distribution used to describe the number of events occurring within a specified observation window. Its single parameter, (\lambda), represents the expected count, and its possible values are the nonnegative integers. In statistics, it is a basic model for count data, particularly when events occur independently at a stable average rate. Unlike a binomial count, a Poisson count has no finite upper bound. (itl.nist.gov)
Definition and probabilities
A random variable (X) has a Poisson distribution with parameter (\lambda>0), written (X\sim\operatorname{Poisson}(\lambda)), if its probability mass function is
[ P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}, \qquad k=0,1,2,\ldots. ]
Here (k!) denotes the factorial of (k), with (0!=1). The probabilities sum to one because the exponential series gives (\sum_{k=0}^{\infty}\lambda^k/k!=e^\lambda). The limiting case (\lambda=0) is usually included by defining (X=0) with probability one. (online.stat.psu.edu)
Its cumulative distribution function, for (x\geq0), is
[ F(x)=e^{-\lambda} \sum_{k=0}^{\lfloor x\rfloor}\frac{\lambda^k}{k!}, ]
and (F(x)=0) for (x<0). Thus cumulative probabilities form a step function rather than a smooth curve. (itl.nist.gov)
For an illustrative model with three arrivals expected per hour, the probability of exactly two arrivals in one hour is (e^{-3}3^2/2!\approx0.2240). The probability of no arrivals is (e^{-3}\approx0.0498). These are consequences of the assumed model, not evidence that any particular arrival system follows it.
Moments and shape
The expected value and variance are identical:
[ E[X]=\lambda,\qquad \operatorname{Var}(X)=\lambda. ]
Consequently, the standard deviation is (\sqrt{\lambda}). This equality of mean and variance is called equidispersion. The distribution is strongly right-skewed for small (\lambda); its skewness is (1/\sqrt{\lambda}), so it becomes more nearly symmetric as the parameter increases. (online.stat.psu.edu)
For noninteger (\lambda), the most probable count is (\lfloor\lambda\rfloor). For positive integer (\lambda), both (\lambda-1) and (\lambda) are modes. These statements follow from the adjacent-probability ratio
[ \frac{P(X=k+1)}{P(X=k)} =\frac{\lambda}{k+1}. ]
The mean need not be an integer even though every observed count is. (itl.nist.gov)
Relation to the binomial distribution
The Poisson distribution arises as a limit of the binomial distribution. If (X_n\sim\operatorname{Binomial}(n,p_n)), with (n\to\infty), (p_n\to0), and (np_n\to\lambda), then, for every fixed nonnegative integer (k),
[ P(X_n=k)\longrightarrow e^{-\lambda}\frac{\lambda^k}{k!}. ]
This explains its association with numerous opportunities for individually unlikely events. Each opportunity can be represented by a Bernoulli variable, while their total counts the successes. (online.stat.psu.edu)
For finite (n), a Poisson distribution with (\lambda=np) can approximate binomial probabilities when (n) is large and (p) is small. The distinction remains important: a binomial variable cannot exceed (n), whereas the Poisson approximation assigns positive probability to all nonnegative counts. (online.stat.psu.edu)
Poisson processes and aggregation
A Poisson process is a stochastic process describing event occurrences over time. In its homogeneous form, counts in disjoint intervals exhibit statistical independence, and the distribution of an interval count depends only on the interval’s length. For rate (r),
[ N(t)\sim\operatorname{Poisson}(rt). ]
The rate (r), measured in events per unit time, differs from the dimensionless expected count (\lambda=rt). Successive waiting times follow an exponential distribution with mean (1/r). The distribution describes a count; the process describes an entire sequence of occurrences. (probabilitycourse.com)
Independent Poisson counts are closed under addition: if (X_i\sim\operatorname{Poisson}(\lambda_i)), their sum is Poisson with parameter (\sum_i\lambda_i). Similarly, independently combining Poisson point processes adds their intensity measures. Independently retaining events with a fixed probability produces another Poisson process, an operation known as thinning. These properties support models involving combined or partially observed event streams. (stat.berkeley.edu)
For large (\lambda), a normal distribution with mean and variance (\lambda) provides an approximation. This relationship can be understood through the central limit theorem and the addition property. A continuity correction, such as replacing (P(X\leq k)) by a normal probability up to (k+\tfrac12), improves the correspondence between discrete counts and continuous values. (online.stat.psu.edu)
Estimation, regression, and limitations
For independent observations (x_1,\ldots,x_n) sharing one parameter, the likelihood function is
[ L(\lambda)= \frac{e^{-n\lambda}\lambda^{\sum_i x_i}} {\prod_i x_i!}. ]
Maximum likelihood estimation yields the sample mean, (\widehat{\lambda}=\sum_i x_i/n). If all counts are zero, the estimate is zero when that boundary value is included in the parameter space. (statproofbook.github.io)
Poisson regression extends the distribution to counts whose conditional means depend on explanatory variables. It is a generalized linear model, commonly using a logarithmic link. An exposure term allows counts observed over different durations or population sizes to be modeled as rates. Examples include customer-service calls and traffic counts. (online.stat.psu.edu)
A principal limitation is the prescribed conditional equality of mean and variance. Overdispersion occurs when variability exceeds the Poisson model’s prediction, potentially because observations differ in unmodeled ways. Quasi-Poisson models relax the variance relationship; negative binomial models provide another distributional alternative. Comparing variance and mean must account for differences in modeled conditional means, rather than relying only on pooled counts. (online.stat.psu.edu)