Suppose you have a set of resistors R, all of which are distributed with mean μ and variance σ.

Consider a section of a circuit with the following layout: (r) || (r+r) || (r+r+r). The equivalent resistance of each part is r, 2r, and 3r. The variance of each section would then be $σ^2$, $2σ^2$, $3σ^2$.

What is the variance in the resistance of the entire circuit?

After sampling several million points, we found that the variance is approximately $.10286sigma^2$.

How would we arrive to this conclusion analytically?

Edit: Resistance values are assumed to be normally distributed with some mean resistance r and variance $σ^2$.

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#### Best Answer

The equivalent resistance $R$ of the entire circuit solves $$ frac1R=sum_{i=1}^{3}frac{1}{R_i}. $$ One assumes that $R_i=imu+sigmasqrt{i}Z_i$, for some independent random variables $Z_i$, centered and with variance $1$.

Without further indications, one cannot compute the variance of $R$, hence, to go further, we consider the regime where $$ color{red}{sigmallmu}. $$ Then, $$ frac{1}{R_i}=frac1{imu}-frac{sigma}{mu^2}frac{Z_i}{isqrt{i}}+text{higher order terms}, $$ hence $$ frac{1}{R}=frac{a}{mu}-frac{sigma}{mu^2}Z+text{higher order terms}, $$ where $$ a=sum_{i=1}^{3}frac1{i}=frac{11}6,qquad Z=sum_{i=1}^{3}frac{Z_i}{isqrt{i}}. $$ One sees that $$ mathrm E(Z)=0,qquadmathrm E(Z^2)=b,qquad b=sumlimits_{i=1}^{3}frac1{i^3}=frac{251}{216}. $$ Furthermore, $$ R=frac{mu}a-frac{sigma}{a^2}Z+text{higher order terms}, $$ Thus, in the limit $sigmato0$, $$ mathrm E(R)approxfrac{mu}a=frac6{11}mu, $$ and $$ text{Var}(R)approxsigma^2cdotfrac{b}{a^4}=sigma^2cdotleft(frac{6}{11}right)^4cdotfrac{251}{216}=sigma^2cdot0.10286ldots $$ These asymptotics of $mathrm E(R)$ and $text{Var}(R)$ can be generalized to any number of resistances in parallel, each being the result of $n_i$ elementary resistances in series, the elementary resistances being independent and each with mean $mu$ and variance $sigma^2$. Then, when $sigmato0$, $$ mathrm E(R)tofrac{mu}a,quad sigma^{-2}text{Var}(R)tofrac{b}{a^4}, $$ where $$ a=sum_ifrac1{n_i},quad b=sum_ifrac1{n_i^3}. $$

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