Translate exponential distribution into normal distribution

Question

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the exponential parameters to work with normal distribution? I was thinking taking the CDF of the exponential, and converting it to the Z-Score of the normal. Is this a feasible approach?

Answer 1

Let $X$ be a random variable with a CFD $F(x)$ having a PDF $f(x)$. Consider the transformation: $$ u=\Phi^{-1}(F(x)),$$ where $\Phi^{-1}$ is the inverse of the standard normal CDF. For the PDF $f_U(u)$ of this function $u=\Phi^{-1}(F(x))$ one has (with $\varphi(u)$ the PDF of the standard normal distribution) under some regularity conditions on $f(x)$: $$ f_U(u)=f(x)\frac{\varphi(u)}{f(x)}=\varphi(u)$$ So $U$ has a standard normal distribution. For an exponential distribution it works.