# Proof of the PDF Projection Theorem

Let

where is any PDF defined on . We show now that is a PDF on , thus

Proof:

The equivalence of the expected values ( vs. in lines 2 and 3) is an application of the change of variables theorem [75]. For example, let be any function defined on . If , then . This can be seen when the expected values are written as the limiting form of the sample mean of a size- sample set as , i.e.

We show now that is a member of .

Proof: Let be drawn from the PDF and let . We now show that the PDF of is indeed . We prove this by showing that the moment generating function (MGF) of is equal to the MGF corresponding to . Let be the joint moment generating function (MGF) of . By definition,

from which we may conclude that the PDF of is .

Another proof of the PPT is available [76].

Baggenstoss 2017-05-19