Problem Definition

Cow	Milk(Y)	Age(X1)	Weight(X2)
#1	10	1	2
#2	11	3	3
#3	12	4	1

We want to find the model which well explains our target variable($y$) with $x$ variables. The model looks like this

$$Y_i =\beta_1X_{1i}+\beta_2X_{2i}+\epsilon_i$$

We can evaluate how precise our model it is with a fluctuation of our error. When we assume that our expected error is zero, the fluctuation represents the size of precision.

• Good for Intuition: $E[|\epsilon-E(\epsilon)|]=E[|\epsilon|]$

• Good for calculation: $\sqrt{E[\epsilon^2]}=\sigma_\epsilon$​

If we make a probabilistic assumption for error, we can easily find the fluctuation. For example, Error can be $-2, -1,0,1,2$ with the probability $\dfrac{1}{5}$. Then $E[|\epsilon|]=1$. However, in a real world problem, we couldn't make a probabilistic assumption for error. Even if we do, we just assume the normal with unknown variance. So to know the precision we need to estimate the sigma of error.

MLE: $\hat{\sigma_\epsilon}=\sqrt{\dfrac{\epsilon_1^2+\cdots+\epsilon_n^2}{n}}$ | $\hat{\sigma_\epsilon}=\sqrt{\dfrac{\epsilon_1^2+\cdots\epsilon^2_{n-factor \;num}}{n-factor\;num}}$