# Random uncertainties

Hi

In http://sl-proj-bi-specification.web...pecification/Activities/Glossary/techglos.pdf it says that: ... if the sources of uncertainties are numerous, the Gaussian distribution is generally a good approximation.

I don't quite understand why. The Central Limit Theorem (CLT) only says that if we have a sum S of N random variables, then S will be Gaussian for very large N. So the CLT does not explain the above. In that case, where does the statement come from?

Best,
Niles.

## Answers and Replies

statdad
Homework Helper
The general interpretation is that the effects of those uncertainties are additive - that's where the CLT comes in.

But that would only explain why the errors are Gaussian, not why the measured variable is Gaussian.

statdad
Homework Helper
If the problem is a location problem, the "model" can be described as

Variable = Mean value + Random error

with "Mean value" a constant. Since the random error is Gaussian, so is the variable.

If the problem is a location problem, the "model" can be described as ...

I am not sure I understand what you mean by "location problem". In my case we are talking about measured speeds.

statdad
Homework Helper
A "location problem" simply means you are trying to determine the mean value of a variable. A mean is one type of measure of location, or measure of center.

I don't know exactly what type of problem you're involved in: my posts above were

1) to show how the Gaussian distribution arises from the "many sources of uncertainty"
2) to show one way in which a measured random quantity can be assumed to have a gaussian distribution

Ok, I understand. In post #2 and #4 you use "error" and "uncertainty" interchangeably. Does

Variable = Mean value + Uncertainty

also hold for a location problem?

Best,
Niles.

statdad
Homework Helper
I guess - the main idea is that the variable is a constant value + some unmeasurable random behavior, which is often modeled by a normal (Gaussian) distribution.

An approach slightly more general than the location model is given by

Variable = Model + Error

where Model'' is some deterministic (non-random) expression. Consider multiple regression:

$$Y = \underbrace{\beta_0 + \beta_1 x_1 + \dots + \beta_p x_p}_{\text{Model}} + \varepsilon$$

with $\varepsilon$ is the random error component

Last edited:
Thanks, it was very kind of you.

Best wishes,
Niles.