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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?

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    $\begingroup$ It probably makes more sense to adapt the formulas so they fit with the data you have than to take the approach of Procrustes and try to mangle the data fit the formulas. You may be able to transform the data to look more or less normal ... but whether that will actually produce meaningful outputs depends intimately on what the formulas do. $\endgroup$
    – Glen_b
    May 28, 2015 at 6:52

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The Z-score is a linear transform$$Z=\frac{X-\mu}{\sigma}$$hence cannot turn an Exponential variable into a Normal variable, even when using the cdf transform $$Z=\frac{F(X)-\mu_F}{\sigma_F}$$ The way to transform an Exponential variate into a Normal variate is to use the cdf followed by the inverse cdf:$$Z=\Phi^{-1}(1-\exp\{-\lambda X\})$$is normally $\text{N}(0,1)$ distributed. However, why would you be interested in a $\text{N}(0,1)$ variate?

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    $\begingroup$ I don't think this is what the OP was describing (which is rather unclear). My interpretation is that the post proposes a probability integral (quantile-to-quantile) transform, exactly as you suggest at the end. In light of @Glen_b's comment, with which I strongly agree, we should be concerned that without any knowledge of the details or actual purposes of these formulas, offering any solution or remarks at this point could badly mislead the OP. $\endgroup$
    – whuber
    May 28, 2015 at 13:36
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    $\begingroup$ @whuber: I cannot but agree! $\endgroup$
    – Xi'an
    May 30, 2015 at 12:55
  • $\begingroup$ @Xi'an, how would you handle $\varphi^{-1}$ in practice? There's no closed-form. $\endgroup$ Jan 22, 2020 at 16:43

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