# Propagation of errors in non-linear function of uncorrelated variables

I have a non-linear recurrence relationship $Y$, where $\beta$ and $\sigma$ are variables from continuous distributions, à priori uncorrelated. Is there anything I can say about error propagation?

\begin{align} Y_i=&\sigma_i+\beta_i(\sigma_{i-1}+\beta_{i-1}(\sigma_{i-2}+...))\nonumber\\ =&\sigma_i+\beta_i\sigma_{i-1}+\beta_i\beta_{i-1}\sigma_{i-2}+...\nonumber\\ =&\sum_{j=0}^i \left(\prod_{k=j+1}^i \beta_k\sigma_j\right) \end{align}

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I don't know if you're doing this from a physics class, but if it's a lab course, don't the experimental fields usually have their own opinion about what constitutes correct error propagation? If you're asking about it as a math question, then maybe there's a rephrasing of your question in terms of mathematical concepts such as standard deviations and such? If you have some mathematically formal theory of "error propagation" you may want to include that. –  Jeff Jun 28 '13 at 21:16
@Jeff This is from a theoretical/conceptual point of view. The population means and standard deviations of the variables are known. The formulae for error propagation involve partial derivatives of the function with respect to the variables: $$\sqrt{\dfrac{\partial Y}{\partial \beta}\sigma_{\beta}+\dfrac{\partial Y}{\partial \sigma}\sigma_{\sigma}}$$. I'm not sure how this applies in this case. Do you think it does? –  HCAI Jun 28 '13 at 21:27