Error propagation

by Nikolai V. Shokhirev

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Introduction

In the section Error estimation we derived that the variance of a sum of independent variables is the sum of variances:

(1)

The individual item can be a function of the other random variables. Each variable X can be presented as its mean value x and a deviation 

Expanding a function in Taylor series and keeping only the first-order terms, we can again reduce each item to a sum. Let as consider important specific cases.

1. Product

A = B · C
	(3)

It can be rewritten as 

(4)

2. Ratio

(5)

However it still can be rewritten as 

(4)

It means that for any combination of products and fractions the variance is the sum of the form (4).  

3. Power

(6)

Eq.(6) can be also presented as

(6a)

For the case of  square root (p = 1/2) we have

(7)

4. Logarithm

(8)

5. Exponent

(9)

You can combine all of the above function and derive new ones in a similar way.

Example

(6)

Eq.(6) can be also presented as

(6a)

For the case of  square root (p = 1/2) we have

(7)

4. Logarithm

(8)

5. Exponent

(9)

You can combine all of the above functions and derive new ones in a similar way.

Example

(10a)

According to (1)

(10b)

Using the above equations (3-9)

(10c)

And finally

(10d)

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