The distribution of the parameters in a two-parameters fit

A visual explanation of why the uncertainties on parameters are found by looking for the intersection of profile of the χ² with a plane crossing it at its minimum + 2.4

Let’s start with a sample of data like the following

We can fit these data with a straight like y=A+Bx. The parameters a and b are found by maximising the likelihood, or minimising the χ². Their uncertainties is found looking for the intersections of the profile of the χ² as a function of each parameter with a line at the minimum χ² + 2.4. Why? Well, the reason is that, in this case, the χ² can be written as a paraboloid, expanding it with Taylor in the vicinity of its minimum up to the second order, such as

When the second derivatives are such that

and similarly for B, the χ² can be rewritten as

The two extra terms represent as many random variables, each distributed as a normal, and squared. This makes the χ² itself, considered as a function of A and B, a random variable that fluctuates as a constant (χ²ₘᵢₙ) plus two normal variables squared, i.e., as a χ² with two degrees of freedom.