# 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:

The denominator is essentially the PVWatts DC model, except the reference cell temperature is not $$T_{\mathrm{STC}}$$ but rather a “typical/average” cell temperature. The report has this to say about that decision:

## 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 $$f_a * f_b$$: