## Coriolis Statistics

### Statistical Relativity and the Coriolis Effect

##### The Coriolis Effect is well known to engineers and provides a parable for the relationship between a sample statistic and the population parameter that it estimates.

*There is a narration so you may want to turn down your computer’s volume. To see full screen (but with only the original resolution), click on the open square symbol.*

At first there may seem little similarity between this well-known Physis 101 demonstration, and the behavior of sample statistics, but the similarities are quite profound.

- The video demonstrates that while the ball’s trajectory is “really” rectilinear (to an observer on the ground), it is obviously curved relative to any measurements made on the revolving merry-go-round.
- As we will show, the behavior of a random variable may “really” be normal (with respect to known population parameters \(\mu \text{ and } \sigma\) (mean and standard deviation), but it is very much different relative to sample statistics \(\bar{X} \text{ and } s\).

Most engineering Monte Carlo studies begin with the model parameters as given. How could it be otherwise?

But in any real situation the model parameters are *not* known. In fact, even the model itself is not known. Normal? lognormal? Weibull? Beta? or mixtures of these? And these are continuous densities. What of the discrete densities? Or non-standard densities? Even when the model can be taken as known, its parameters (e.g. \(\mu \text{ and } \sigma\) for a Normal model) still must be estimated from the data.

Since the true values of \(\mu \text{ and } \sigma\) are not known, engineers “learned” to compute \(\bar{X} \text{ and } s\) and then substitute them for \(\mu \text{ and } \sigma\).

Unfortunately, this isn’t what we were taught, even if it does appear to be what we “learned.” So what’s wrong with this picture?

In practice, statistical model parameters, like \(\mu \text{ and } \sigma\), are not usually known, and must be inferred from \(\bar{X} \text{ and } s\). Random samples from a Normal density often display very non-Normal behavior, especially when the sample size is small (\(n \lt 30\)), tempting the use of another model (*e.g.* Weibull) which may fit the current sample better, but be less effective in predicting future observations.

##### Credits:

Among several other places, I found this video at the PBS website, referenced as part of the University of Illinois WW2010 Project where is is found as http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/crls.rxml