super-Poissonian
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In mathematics, a super-Poissonian distribution is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
that has a larger
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
than a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
with the same
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
. Conversely, a sub-Poissonian distribution has a smaller variance. An example of super-Poissonian distribution is
negative binomial distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
. The
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
is a result of a process where the time (or an equivalent measure) between events has an
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
, representing a
memoryless In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already ...
process.


Mathematical definition

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
it is common to say a distribution, ''D'', is a sub-distribution of another distribution ''E'' if ''D'' 's
moment-generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
, is bounded by ''E'' 's up to a constant. In other words : E_ exp(t X)\le E_ exp(C t X) for some ''C > 0''. This implies that if X_1 and X_2 are both from a sub-E distribution, then so is X_1+X_2. A distribution is ''strictly sub-'' if ''C ≤ 1''. From this definition a distribution, ''D'', is sub-Poissonian if : E_ exp(t X)\le E_ exp(t X)= \exp(\lambda(e^t-1)), for all ''t > 0''. An example of a sub-Poissonian distribution is the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabil ...
, since :E exp(t X)= (1-p)+p e^t \le \exp(p(e^t-1)). Because sub-Poissonianism is preserved by sums, we get that the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
is also sub-Poissonian.


References

Poisson point processes Types of probability distributions {{physics-stub