No σ for you!

[carey]

Albert, All good points!

(11-27-2024 01:09 PM)Albert Chan Wrote:  

(11-25-2024 04:49 PM)carey Wrote:  \[
Gaussian function = f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right),
\]

OT, I find standard normal (µ=0, σ=1) less confusing.
Above f(x) has unit of 1/σ, but pdf(z) is dimensionless.

pdf(z) := exp(-z*z/2) / sqrt(2*pi)      , where z = (x-µ)/σ

1) The fact that the normal distribution has “unit of 1/σ” supports the claim that σ is Nature’s measure (unit of) variability.
2) Interestingly, σ has no units other than the units of the numbers used to calculate it. If numbers only are used, then σ is unitless.

(11-27-2024 01:09 PM)Albert Chan Wrote:  Dimensionless equation is better. More compact, and *only* 1 shape

Yes, there are many benefits of non-dimensionalization…except one: model testing, because models have units! For example, while y=t is a linear function, it doesn’t become a family of one-parameter linear models until we introduce a parameter, e.g., y = A t, and that doesn’t become a specific one-parameter linear model until A is a number (with units), y (m) = 4.02 (m/s) t (s), which can be tested experimentally.