Suppose that $X\sim Exp(p)$ is exponentially distributed with expectation $1/p$. Does there exist transformation $f:\mathbb{R}\to\mathbb{R}$ such that $f(X)$ will be some (ideally "nice") distribution with expectation $\mathbb{E}f(X)=p$? Here, $f$ can not depend on $p$.
Context: In GLM models (generalized linear models) we usually deal with a random variables $Y\sim exp(p)$ where $p=\beta_0+\beta_1X_1$. So $\mathbb{E}Y=\frac{1}{\beta_0+\beta_1X_1}$ which is kind of ugly. It would be nice if we can change Y in such a way that $f(Y)$ will have nice linear expectation. Is it possible?