Numerical Analysis - Rounding after calculating uncertainty of a function

Question

I successfully calculated the uncertainty associated to a group of functions:

• f(x) +- 0,0104
• g(x) +- 0,0155
• h(x) +- 0,5445

The problem is the result of these functions are rounded to the nearest integer number as in: round(f(35)) = round(17,7549) = 18. Considering t(x) = round(h(x)) how can I calculate the uncertainty of t(x)?

Answer 1

One way is to consider the effect of rounding as a uniform random variable $\delta$ (Widrow et al.). Then you model your uncertainty as a random variable $\epsilon$. The result is that $$ t(x) \approx h(x) + \delta + \epsilon = h(x) + \nu$$

If you model $\epsilon$ as a uniform random variable too, then the density total noise $\nu$ will be the convolution of the densities of $\delta$ and $\epsilon$, which is a triangular-shaped density (convolution of two rectangles). From there, you can compute the standard deviation and other quantities of interest.