(calibration_circe_principle_normality_hypothesis)=
# The normality hypothesis

If we collect all the information about the system described so far, the problem can be summarised 
as 

```{math}
R^{\rm exp}_j - R^{\rm code}_j = (R^{\rm exp}_j - R^{\rm real}_j) + (R^{\rm real}_j - R^{\rm code}_j) 
= e_j + \sum_{i=1}^{q}\dfrac{\partial R^{\rm code}_j}{\partial \alpha_i} \times \alpha_{j,i}
```

In this expression, one can discuss the different contributions:

- $R^{\rm exp}_j - R^{\rm code}_j$ and $\dfrac{\partial R^{\rm code}_j}{\partial \alpha_i}$ are known.

- $e_j$ is a realisation of $\mathcal{N}(0,(\sigma^{\rm exp})^{2})$ where $\sigma^{\rm exp}$ is also 
known.

- The $\alpha_{j,i}$ are unknown. The only available information comes from their statistical 
properties: their bias $b_i$ and their standard deviation $\sigma_i$.

Several solutions are possible for the vector $\alpha$, leading to a needed choice among 
them. The criterion chosen to do so is the maximum likelihood estimation, which requires a hypothesis
on the form of the probability distribution followed by the $\alpha_i$ parameters. The normality assumption is then adopted.