Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function

Question

I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows: $$ Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,. $$ Then, I need to take Laplace-Stieltjes from Y as follows: $$ W = \int_0^\infty se^{-st}Y(t)dt\, =\int_0^\infty se^{-st} \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\ dt\ $$ I think you can read the expression and help with the question.

Actually, in above-mention problem z parameter have a special bounds as follows:

0 < z < 1

For example consider z = 0.5; Is there any solution or approximation for this problem at this condition?

Answer 1

A method of solving is shown below :