So the distribution of

is geometric.

But the distribution of

is not geometric, right?

Mr F says: Correct. The distribution of is not geometric. It has a negative binomial distribution.
I am having trouble determining the distribution of

. My original thought was that it was perhaps the exponential distribution because the exponential distribution can be considered the continuous version of the geometric distribution, and unless I have made an error in my calculations, the Laplace Transform of the exponential dist. is not the same as the solution given in the question.

Mr F says: is clearly a discrete random variable and therefore cannot have a continuous distribution.
Just to double check though, the Laplace transform of a continuous random variable X is

where

is the pdf of

??