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I have a question regarding Monte Carlo simulation (direct simulation), applied to propagation of uncertainties.

From what I understand Monte Carlo accepts random numbers of each input variable of the model.

These random numbers are generated with the mean, standard deviation and type of PDF (normal, triangular, uniform, etc.)

But the question is: I have 2 input variables in my model:

x: mean=30.5 std=0.001 <- calculated with 95% confidence.
y: mean=2 std=0.1 <- calculated with 99% confidence

with z = x*y

What confidence interval will z be generated? How do I know which uncertainty (standard deviation) of z is correspondent to a 95% confidence level?

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    $\begingroup$ I cannot make sense of this question for several reasons and hope you might be able to clarify them. What does it mean to "calculate" an input variable "with 95% confidence"? What do you mean by a "confidence interval" for x*y? What do you mean for a standard deviation to "correspond" to a confidence level? $\endgroup$ – whuber Aug 30 '13 at 15:51
  • $\begingroup$ Ok. So i work with equipment's that have specifications or uncertainties. Those uncertainties are generally at 95% confidence level (some at 99%). To calculate the final uncertainty (ex: Power = Voltage*Current), i have to combine the uncertainty of the equipment generating voltage and current, so the final uncertainty of the power produced is a combination of both uncertainty of voltage and current. Generally the GUM method is applied but Monte Carlo is also accepted. $\endgroup$ – humberto Aug 30 '13 at 15:57
  • $\begingroup$ The Gum strips the (expanded)uncertainties and work usually at sigma=1, only in the end we expand the uncertainty for the confidence level chosen. But here i don't know what implications i have mixing different uncertainties produced with different confidence levels. $\endgroup$ – humberto Aug 30 '13 at 16:00
  • $\begingroup$ And since degrees of freedom are not calculated, i cannot expand the uncertainty. I don't really know if this is correct or not but that's why i'm here. $\endgroup$ – humberto Aug 30 '13 at 16:03
  • $\begingroup$ Also. The distributions (PDF) are known. $\endgroup$ – humberto Aug 30 '13 at 16:05
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Finish 10.000 MC runs and then start computing your confidence intervals. Compute e.g. the median value, which divides your probability distribution (PD) into two parts, where each part corresponds to 50% probability or area of your PD. Integrate your PD from -infinity to the z value covering 2.5% area of your PD, and integrate from zero to the z value covering 97.5% area of your PD. In total you then have the z values, which enclose 95% of your PD. This is denoted the 95 percentile. You can then compute something like a standard deviation by: z1=z975-median and z2=median-z025. Maybe have a look at "confidence.pro" (IDL language).

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