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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