# Difference in Probability Measure vs. Probability Distribution

I am trying to better understand the Difference in “Probability Measure” and “Probability Distribution

I came across the following link : https://math.stackexchange.com/questions/1073744/distinguishing-probability-measure-function-and-distribution

” The difference between the terms “probability measure” and “probability distribution” is in some ways more of a difference in connotation of the terms rather than a difference between the things that the terms refer to. It’s more about the way the terms are used. “

The answer over here suggests that these two concepts might be the same thing?

In this case – could we consider the “Normal Probability Distribution Function” as a “Probability Measure”?

Thanks!

First off, I am not used to the term “Probability Distribution Function”. In case you are referring to a “PDF” when you are using “Probability Distribution Function”, then I would like to point out that PDF is rather the abbreviation for “Probability Density Function”. Below, I will presume you meant probability density function.

Second, the word “distribution” is used very differently by different people, but I will refer to the definition of this notion in the scientific community.

In a nutshell: The distribution of a random variable $$XX$$ is a measure on $$R\mathbb{R}$$, while the PDF of $$XX$$ is a function on $$R\mathbb{R}$$ and the PDF doesn’t even always exist. So they are very different.

And now we get to the mathematical details:
First, let’s define the term random variable because that is what all those terms usually refer to (unless you get a step further and want to talk about random vectors or random elements, but I will restrict this here to random variables). I.e., you talk about the distribution of a random variable.

Given a probability space $$(Ω,F,p)(\Omega, \cal{F}, p)$$ ($$Ω\Omega$$ is just a set, $$F\cal{F}$$ is a sigma algebra on $$Ω\Omega$$, and $$pp$$ is a measure on $$(Ω,F)(\Omega, \cal{F})$$), a random variable $$XX$$ is a measureable map $$X:Ω→RX: \Omega \to \mathbb{R}$$.

Then we can define: The distribution $$pXp_X$$ of the random variable $$XX$$ is the measure $$pX=p∘X−1p_X = p \circ X^{-1}$$ on $$R.\mathbb{R}.$$

I.e. you push the measure $$pp$$ from $$Ω\Omega$$ forward to $$R\mathbb{R}$$ via the measurable function $$XX$$.

Next we define the probability density function (PDF) of a random variable $$XX$$: The PDF $$fXf_X$$ of a random variable $$XX$$, if it exists, is the Radon-Nikodym derivative of its distribution w.r.t. the Lebesgue measure $$λ\lambda$$, i.e. $$fX=dpXdλf_X = \frac{d\,p_X}{d\,\lambda}$$.

So the distribution $$pXp_X$$ and the PDF $$fXf_X$$ of a random variable are very different entities (to a stickler, at least). But very often, if the PDF $$fXf_X$$ exists, it contains all of the relevant information about the distribution $$pXp_X$$ and can thus be used as a handy substitute for the unwieldy $$pXp_X$$.