# Posts tagged math

## T_cell_typ_avg

This is a short note that derives the “typical average cell temperature” used in the “Weather-Corrected Performance Ratio”. The Weather-Corrected Performance Ratio 1 is defined as:

 \mathrm{PR_{corr}} = \frac{\sum_i \mathrm{EN_{AC}}{_i}} {\sum_i \left[ P{\mathrm{STC}} \left( \frac{G_{\mathrm{POA}_i}}{G_{\mathrm{STC}}}\right) \left( 1 - \frac{\delta}{100}(T_{\mathrm{cell_typ_avg}} - T_{\mathrm{cell}_i})\right) \right] }

## Summing Uniform Distributions

The distribution of the sum of N uniformly distributed variables came up in a recent RdTools PR. The approach taken there was to just sample each distribution and sum the samples, but I wondered how it could be done analytically. This notebook is inspired by this StackExchange post and extends the derivation from just summing two uniform distributions on $[0,1]$ to two distributions with arbitrary bounds.

Given two independent random variables $A$ and $B$ with probability densities $f_a(x)$ and $f_b(x)$, the probability density $f_c(x)$ of their sum $A + B = C$ is given by the convolution $f_a * f_b$: 