# Random uncertainties

1. Sep 17, 2010

### Niles

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.

2. Sep 17, 2010

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

3. Sep 17, 2010

### Niles

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

4. Sep 17, 2010

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.

5. Sep 17, 2010

### Niles

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

6. Sep 17, 2010

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

7. Sep 17, 2010

### Niles

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.

8. Sep 17, 2010

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: Sep 17, 2010
9. Sep 17, 2010

### Niles

Thanks, it was very kind of you.

Best wishes,
Niles.