Definition 3.9. Let be a nonnegative random variable with density function . The Laplace-transform is defined as

Theorem 3.10. The Laplace-transform holds the following properties

Proof.

1.

2. The lower bound of comes from the observation that is nonnegative and thus the integral is also nonnegative. For the upper bound we have

3. because if are independent then , . are also independent and hence the multiplicative law is valid.

4.

hence .

The main advantage of the Laplace-transform is that it can be used to solve differential equations. It should be noted that the Laplace-transform can be applied for any function with nonnegative range. In the following we would like to solve some differential equations that is why we need.

Theorem 3.11. The Laplace-transform hold the following properties

1.

2.

Proof. 1.

2. Using integration by parts

Theorem 3.12. Laplace-transform of a random sum

Proof. By the theorem of total expectation we have

Due to their practical importance we state the following theorems without proof.

Theorem 3.13. yields the following limits

1. Initial value theorem

2. Final value theorem

Theorem 3.14. (POST-WIDDER inversion formula) If is a continuous and bounded function on , then

Theorem 3.15. (Continuity Theorem) Let be a sequence of random variables having distribution functions . If , where is the distribution function of some random variable then for the corresponding Laplace-transform we have

and conversely

if the sequence of Laplace-transforms converges to a function then for the corresponding distribution functions we have

and the limiting function is the Laplace-transform of some random variable distribution function .

In the following let us find the Laplace-transform of some important distributions.

Example 3.8. Find the Laplace-transform if .

Solution:

Example 3.9. Find the Laplace-transform if .

Solution:

Since is the sum of independent exponentially distributed random variables with the same parameter by applying the convolution property of the Laplace-transform we have

Example 3.10. Find the Laplace-transform of a hypoexponential distribution.

Solution:

Since a hypoexponentially distributed random variable is the sum of independent exponentially distributed random variable with different parameters we obtain

In the next example we show how the th moment of an exponentially distributed random variable can be obtained in a simple way by using one of the properties of the Laplace-transform. This calculation is rather cumbersome by the density function approach.

Example 3.11. Using the Laplace-transform show that if , then

Solution:

Example 3.12. Let be a geometrically distributed counting random variable and summands. Find the distribution of the random sum.

Solution:

Knowing that if , then , thus

That is exactly the Laplace-transform of .

Theorem 3.16. Laplace-transform of a mixture distribution is the mixture of the corresponding Laplace-transforms.

Proof. Let

Then

Example 3.13. Find the Laplace-transform of the function .

Solution:

Example 3.14. Use the Laplace-transform to solve the system of differential equations

with initial conditions

Solution:

By taking the Laplace-transform of both sides we have

Using integration by parts we get

Assuming that is bounded that is then

Consequently

Using the initial condition we obtain

and

After substitution we get

thus

Furthermore

hence

and thus it is easy to see that

Keeping in mind the previous example finally we get the solution as