An Inverse Gaussian Distribution is a continuous probability distribution that is used to model data that is positively skewed and has a long tail. It is also known as the Wald Distribution and is often used in statistical analysis to model the time between events.
An Inverse Gaussian Distribution is characterized by two parameters: the mean, denoted by μ, and the shape parameter, denoted by λ. The distribution is unimodal, meaning it has only one peak, and is positively skewed. It is also a continuous distribution, meaning it can take on any value within a certain range.
The Inverse Gaussian Distribution is closely related to the Normal Distribution. In fact, it can be thought of as a generalization of the Normal Distribution. When the shape parameter λ is large, the Inverse Gaussian Distribution closely resembles the Normal Distribution. However, as λ decreases, the distribution becomes more skewed and has a longer tail.
The Inverse Gaussian Distribution has many real-life applications, particularly in the fields of finance and insurance. It is often used to model the time between financial events, such as stock price changes or loan defaults. It is also used in insurance to model the time between claims.
The probability density function (PDF) of the Inverse Gaussian Distribution is given by the formula f(x; μ, λ) = (√(λ/2πx^3)) * exp(-(λ(x-μ)^2)/2μ^2x). To calculate the probability of a specific value or range of values, this formula can be integrated. However, most statistical software packages have built-in functions for calculating probabilities and other statistics related to the Inverse Gaussian Distribution.