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 ...
, the law of rare events or Poisson limit theorem states that 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 ...
may be used as an approximation to the
binomial distribution, under certain conditions.
The theorem was named after
Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
(1781–1840). A generalization of this theorem is
Le Cam's theorem In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following.
Suppose:
* X_1, X_2, X_3, \ldots are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), n ...
.
Theorem
Let
be a sequence of real numbers in
such that the sequence
converges to a finite limit
. Then:
:
Proofs
:
.
Since
:
and
:
This leaves
:
Alternative proof
Using
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
, we can write:
:
Letting
and
:
:
As
,
so:
:
Ordinary generating functions
It is also possible to demonstrate the theorem through the use of
ordinary generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
s of the binomial distribution:
:
by virtue of the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Taking the limit
while keeping the product
constant, we find
:
which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
.)
See also
*
De Moivre–Laplace theorem
In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particu ...
*
Le Cam's theorem In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following.
Suppose:
* X_1, X_2, X_3, \ldots are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), n ...
References
{{Reflist
Articles containing proofs
Probability theorems