Why is the Laplace tranform of the pdf of a random variable called the Laplace transform of that variable?

Question

We know that moment generating function of a random variable $X$ is $$M(t)=E[e^{tX}]$$ and if we replace $t$ with $-s$ then we get the Laplace transform as follows: $$\int_{-\infty}^{\infty}e^{-sx}f(x)dx$$ Now this is actually the Laplace transform of the pdf of the random variable $X$. Sometimes (actually in my observation every time) this is called the Laplace transform of the random variable. I want to know why it is said so when it is actually the Laplace transform of the pdf of that random variable? Thanks in advance.

Answer 1

It is nothing profound.

"The Lapace transform of the probability density function for a random variable" is way too many words to keep saying and writing.

"The Lapace transform of the random variable" is a bit quicker to say.

Mathematicians are lazy.

That is okay as long as the true meaning is understood.   Although things like this do cause confusion when first encountered.

That is all.