Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Compound Poisson distribution
Summary
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution.
Suppose that
i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that
are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of i.i.d. random variables
is a compound Poisson distribution.
In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.
The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, and this joint distribution can be obtained by combining the conditional distribution Y | N with the marginal distribution of N.
The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus
Then, since E(N) = Var(N) if N is Poisson-distributed, these formulae can be reduced to
The probability distribution of Y can be determined in terms of characteristic functions:
and hence, using the probability-generating function of the Poisson distribution, we have
An alternative approach is via cumulant generating functions:
Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X1.
It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions. And compound Poisson distributions is infinitely divisible by the definition.
When are positive integer-valued i.i.d random variables with , then this compound Poisson distribution is named discrete compound Poisson distribution (or stuttering-Poisson distribution) .
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (12)
MATH-231: Probability and statistics I
Introduction to notions of probability and basic statistics.
CS-433: Machine learning
Machine learning methods are becoming increasingly central in many sciences and applications. In this course, fundamental principles and methods of machine learning will be introduced, analyzed and pr
MGT-499: Statistics and data science
This class provides a hands-on introduction to statistics and data science, with a focus on causal inference, applications to sustainability issues using Python, and dissemination of scientific result
Related publications (67)
Related people (15)
Related concepts (9)
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 constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
Probability-generating function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
Infinite divisibility (probability)
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables Xn1, .