# The definition of amplitude probability density

I'm trying to figure out the formal definition of "amplitude probability density"(APD). First of all I didn't find a textbook which defines APD but there are some sources that explain it intuitively like this:

I first encountered APD when reading about Woodward's theorem. This is taken from "Communication Circuits: Analysis and Design" by Clarke & Hess: By looking at this, it seems to me that we can find APD by finding the first-order pdf of $$f(t, X)$$ for some $$t$$ and $$X$$ is an uniform random variable but the problem still remains. What's the formal definition of APD? If we completely know $$f(t)$$, there is no probability to calculate. So it should be a realization of a random process. Also I found another source which talks about pdf of a signal: So maybe the word "probability" is totally misleading and APD is just amplitude distribution of a known signal? In that case, what's the formal definition of the amplitude distribution?

• I’ve never heard the term before. Is it not just the probability density function of the amplitude of a signal? The pdf of a signal represents the probability of choosing any specific value when you choose randomly.
– Ryan
Aug 18 '21 at 2:51
• @Ryan Thanks. Do you have a formal definition for the probability density function of the amplitude of a signal? I couldn't find a formal definition. Aug 18 '21 at 23:36
• It seems like you know what a pdf is. That’s all it is, any value in the signal can be considered a random variable and therefore it has a pdf. The formal definition is the formal definition of a pdf.
– Ryan
Aug 19 '21 at 1:50