Distribution and density functions of a linear transformation - two short derivations to read during breakfast.

### Distribution function of a linear transformation

Let

The answer is obtained in one line if you know the definition of the distribution function:

(1)

For the inequalities

### Density function of a linear transformation

As above,

Recall formula (1) that links distribution and density functions. Equation (1) in terms of densities becomes

Let's differentiate this equation. The Newton-Leibnitz formula applied to the integral on the left gives

(2)

Equation (2) will be used to derive ML estimators for the linear model.

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