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:
(1) .
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
(2) .
Equation (2) will be used to derive ML estimators for the linear model.
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