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Using a simple model, I have produced an output distribution as

rv = scipy.stats.johnsonsu(1.874, 2.324, 52.633, 1.097)

If this is a manufacturing setting, and my lower spec limit is 45 and upper spec limit is 55, I want to know the expected failure rates (x < 45 or x > 55). I can check the cumulative density function and the survival function, respectively, to perform each of the single tailed tests:

>>> rv.cdf(45)
1.035e-05
>>> rv.sf(55)
3.564e-08

Since this random variable represents an independent draw during manufacturing, if I assemble 1e6 units, should I expect none of them will be larger than 55 but 10 will be less than 45?

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  • $\begingroup$ Why are you doing this? Where did you get those parameters? $\endgroup$ – eric_kernfeld Aug 21 '18 at 15:56
  • $\begingroup$ I want to understand the expected failure rates for the given distribution and the upper/lower limits. Those parameters came from fitting that distribution to a histogram of data. That data was produced by doing a monte carlo simulation of a simple polynomial model with random variable inputs. $\endgroup$ – pixels Aug 22 '18 at 14:26
  • $\begingroup$ Well, I think you're doing it right. $\endgroup$ – eric_kernfeld Aug 22 '18 at 17:40
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If you believe your fitted distribution has negligible error, then you're right. You could be more precise about the phrasing: the number less than 45 is approximately Poisson with expectation about 10.

If you have no guarantee that your fitted distribution is correct, then you could be completely wrong, especially if your fitting procedure focuses on the bulk of the data at the expense of accuracy in the tails. An alternative would be to retain only extreme observations and fit some type of extreme value distribution to them. This would neglect potentially useful information from more typical samples, but it would be easier to understand the uncertainty in your estimates if you focus on how many examples you have of extreme behavior (or how few).

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