by Nikolai V. Shokhirev
Error: Estimation | Amplification | Propagation
In the section Error estimation we derived that the variance of a sum of independent variables is the sum of variances:
(1) |
The individual terms can be functions of the other random variables. Each variable X can be presented as its mean value x and a deviation
X = x + δx | (2) |
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 can still be rewritten as
(4) |
This 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 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) |
©Nikolai V. Shokhirev, 2001-2004