# Cumulitive hazard function and standard exponential distribution

I have the following hw problem,

Assume that the cumulative hazard function $\Lambda_0$ is continuous and strictly increasing on $[0,\infty)$ and denote its inverse by $\Lambda_0^{-1}$. Let $V$ be a standard exponentially distributed random variable. Show that the random variable $\Lambda_0^{-1}(V)$ is distributed according to the distribution with cumulative hazard $\Lambda_0$.

I have difficulties starting because I don't know how to get around the fact that $\Lambda_0$ is not given. In the preceeding problems, I didn't have much trouble deriving the hazard functions, cumulative hazard functions, mean residual life functions, etc., but here the approach is different and I don't even know where to begin.

You have $P(\Lambda^{-1}_0(V) \leq x)=P(V \leq \Lambda_0(x))=1-\exp(-\Lambda_0(x))$. (First step is due to inverse relationship and the second step just plugs in the standard exponential formula) So you end up with an expression for the cdf. Finally, use the general relationship between the cumulative hazard function and the cdf.