A Note on Laplace Transforms of Some Common Distributions Used in Counting Processes Analysis

An important problem of actuarial risk management is the calculation of the probability of ruin. Using probability theory and the definition of the Laplace transform one obtains expressions, in the classical risk model, for survival probabilities in a finite time horizon. Then explicit solutions are found with the inversion of the double Laplace transform; using algebra, the Laplace complex inversion formula and Matlab, for the exponential claim amount distribution.


Introduction
This paper seeks to derive Laplace transforms of common distributions used in counting processes analysis. In mathematics and with many applications in physics and engineering and throughout the sciences, the Laplace transform is a widely used integral transform. Denoted by Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.  [5].
By definition the Laplace transform of a distribution of density function f defined on  or on a part of  is the function defined on the set  of complex numbers by (e.g. see [4] [5] [6] and [7]), The rest of the paper proceeds as follows: Section 2 is related to the main results derived in the paper. Section 3 concludes the paper.

Exponential Distribution and Its Laplace Transform
Proposition 1. The Laplace transform of the exponential distribution, Proof of Proposition 1: The Laplace transform of the law of X is defined as,  The Laplace transform is by definition,

Normal Distribution and Its Laplace Transform
Proposition 3. The Laplace transform of the normal distribution with known parameters µ and 2 σ , and with a density function given by where Φ is the cumulative distribution of the standard normal distribution. Proof of Proposition 3: The Laplace transform is by definition, Φ is the cumulative density function of the standard normal distribution. ■

Inverse Gaussian and Its Laplace Transform
Proposition 4. The Laplace transform of the Inverse Gaussian distribution with parameters λ , µ and with a density function defined by where z is a complex number such that: is a real positive number (e.g. see [7] [8] [9] and [10]).

Pareto Distribution and Its Laplace Transform
with x positive; is given by: where X E is the mathematical expectation of a normal random distribution truncated at 0 and with parameters σ and µ .

Proof of Proposition 8:
The Laplace is by definition, where X E is the mathematical expectation of a normal random distribution truncated below at 0 and with parameters σ and µ . ■

Log-Logistic Distribution and Its Laplace Transform
Proposition 9. The Laplace transform of the log-logistic distribution with pa- where X E is the mathematical expectation of a log-logistic random distribution with parameters α and λ .
Proof of Proposition 9: The Laplace transform is by definition, where X E is the mathematical expectation of a log-logistic random distribution with parameters α and λ . ■

Conclusion
In this paper one derived the Laplace transform of some important distributions used in counting processes. Results have important implications in 1) solving differential equations; 2) solving partial derivative equations; 3) deriving complex impedances; 4) solving partial fraction expansions; 5) conducting convolution analysis; 6) running complex geometric analyses; 7) determining structure of astronomical objects etc.