Super-Poissonian Distribution
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In mathematics, a super-Poissonian distribution is a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
that has a larger
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
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 if these events occur with a known const ...
with the same
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
. 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, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...
. 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 if these events occur with a known const ...
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 or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
, representing a
memoryless In probability and statistics, memorylessness is a property of probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential d ...
process.


Mathematical definition

In
probability theory Probability theory or probability calculus 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 expre ...
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, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...
, 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 and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
is also sub-Poissonian.


References

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