Distribution and density functions of a linear transformation - two short derivations to read during breakfast.
Distribution function of a linear transformation
Let be a random variable and let be its linear transformation (here are some real numbers and , otherwise is not random). If the distribution function is known, what will be the distribution function of ?
The answer is obtained in one line if you know the definition of the distribution function:
For the inequalities and to be equivalent, we have to assume that (for applications this is enough). The case is left as an exercise.
Density function of a linear transformation
As above, is a linear transformation of . Suppose they have densities . What is the relationship between the densities?
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 evaluated at . On the right, additionally, we have to use the chain rule. The result is
Equation (2) will be used to derive ML estimators for the linear model.