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 ...
, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of
continuous 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 ...
s with
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
on (0,∞).
Its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
is given by
:
for ''x'' > 0, where
is the mean and
is the shape parameter.
The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a
Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a fixed positive level.
Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable.
To indicate that a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' is inverse Gaussian-distributed with mean μ and shape parameter λ we write
.
Properties
Single parameter form
The probability density function (pdf) of the inverse Gaussian distribution has a single parameter form given by
:
In this form, the mean and variance of the distribution are equal,
Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is related to the standard normal distribution by
:
where
,
and the
is the cdf of standard normal distribution. The variables
and
are related to each other by the identity
In the single parameter form, the MGF simplifies to
:
An inverse Gaussian distribution in double parameter form
can be transformed into a single parameter form
by appropriate scaling
where
The standard form of inverse Gaussian distribution is
:
Summation
If ''X''
''i'' has an
distribution for ''i'' = 1, 2, ..., ''n''
and all ''X''
''i'' are
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
, then
:
Note that
:
is constant for all ''i''. This is a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for the summation. Otherwise ''S'' would not be Inverse Gaussian distributed.
Scaling
For any ''t'' > 0 it holds that
:
Exponential family
The inverse Gaussian distribution is a two-parameter
exponential family
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
with
natural parameters
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculat ...
−''λ''/(2''μ''
2) and −''λ''/2, and
natural statistics
In theory of probability, probability and statistics, an exponential family is a parametric model, parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, includin ...
''X'' and 1/''X''.
Relationship with Brownian motion
Let the
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
''X''
''t'' be given by
:
:
where ''W''
''t'' is a standard
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. That is, ''X''
''t'' is a Brownian motion with drift
.
Then the
first passage time
Events are often triggered when a stochastic or random process first encounters a threshold. The threshold can be a barrier, boundary or specified state of a system. The amount of time required for a stochastic process, starting from some initial ...
for a fixed level
by ''X''
''t'' is distributed according to an inverse-Gaussian:
:
i.e
:
(cf. Schrödinger equation 19, Smoluchowski, equation 8, and Folks, equation 1).
Suppose that we have a Brownian motion
with drift
defined by:
:
And suppose that we wish to find the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
for the time when the process first hits some barrier
- known as the first passage time. The
Fokker-Planck equation describing the evolution of the probability distribution
is:
:
where
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. This is a
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
(BVP) with a single absorbing boundary condition
, which may be solved using the
method of images The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a resul ...
. Based on the initial condition, the
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
to the Fokker-Planck equation, denoted by
, is:
:
Define a point
, such that
. This will allow the original and mirror solutions to cancel out exactly at the barrier at each instant in time. This implies that the initial condition should be augmented to become:
:
where
is a constant. Due to the linearity of the BVP, the solution to the Fokker-Planck equation with this initial condition is:
:
Now we must determine the value of
. The fully absorbing boundary condition implies that:
:
At
, we have that
. Substituting this back into the above equation, we find that:
:
Therefore, the full solution to the BVP is:
:
Now that we have the full probability density function, we are ready to find the first passage time distribution
. The simplest route is to first compute the
survival function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time.
The survival function is also known as the survivor function
or reliability function.
The term ...
, which is defined as:
:
where
is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of the standard
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The survival function gives us the probability that the Brownian motion process has not crossed the barrier
at some time
. Finally, the first passage time distribution
is obtained from the identity:
:
Assuming that
, the first passage time follows an inverse Gaussian distribution:
: