I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.

Let $x$ and $y$ be positive reals satisfying all the following conditions:

- $f(x,y)$ is strictly increasing for $x \ge 0$
- $f(0,y) = 0$
- $f(1,y) = 1$
- $f(x,y) \to \infty$ as $x \to \infty$
- For every $y > 0$, $$ \int_{0}^{\infty} e^{-y f(x,y)}dx = 1. $$

My questions are:

Is there a exist a function satisfying the all above conditions?

Are there infinitely many functions satisfying the all above conditions?

Preferably I would like to avoid artificially constructed function like piece-wise models. However this is not a strict constraint and if non-piecewise solutions are not available then piecewise solutions will do.

Condition 1-4 are easy to satisfy but condition 5 has made the solution elusive for me. However for my actual problem, I can do with a weaker form of condition 5 which is for all $0 < y < 5$, we must have $\int_{0}^{\infty} e^{-y f(x,y)}dx = 1$.