# Is a probability density function always well-defined?

I’m studying mathematical statistics.

I learned about the concepts of sample space, $$\sigma$$-field, probability set function with probability axioms, random variables, and probability density function.

Let $$S$$ be a sample space. Let $$A$$ be a sigma field. Let $$\mathbb{P}$$ be a probability set function on $$A$$. And let $$X$$ be a random variable on $$S$$.

Then we have a new sample space $$X(S)$$. And its power set is a new sigma field $$\mathcal{F}$$. So, now we can define a function $$\mathbb{P}_X$$ from $$\mathcal{F}$$ to $$\Bbb{R}$$ by $$\mathbb{P}_X(B) = \mathbb{P}_X[X^{-1} (B)]$$ for all $$B \in \mathcal{F}$$.

In my book, the probability density function is defined in that way. And it says this probability density function $$\mathbb{P}_X$$ is always a probability set function. The book says it is an exercise. But I don’t agree with this. I think probability density function is a probability set function only if it is well-defined!!

In some cases, it might be possible that $$X^{-1} (B)$$ is not in $$A$$, which means $$\mathbb{P}_X$$ is not well-defined since $$\mathbb{P}[X^-1(B)]$$ is not well-defined.

So I think $$\mathbb{P}_X$$ is not always well-defined.

• Your $\mathsf{Px}: \mathcal{F}\to \mathbb{R}$ does not look like a probability density function, but a probability measure. To get a density, you need a Radon–Nikodym derivative, and this does not always exist Jun 4, 2018 at 8:15
• I have never seen anyone using the term probability density function for $P_X$. I am almost sure you are mis-quoting the text. Secondly $P_X$ is not defined on the power set of the range of $X$. It is defined on those sets $B$ for which $X^{-1}(B)$ belongs to the original sigma algebra. If you are quoting the definitions as given in the text then throw that book away immediately. Jun 4, 2018 at 8:15
• Oh sorry, yes you are right. I’m mis-quoting it. It says small px is a pdf. It is defined on $\mathsf{X(S)}$ and, for each $x_{k} \in X(S)$, px($x_k$) = Px[{$x_k$}]. And the book deals with the discrete case. So book says Px satisfies probability axioms and px is a pdf. But my question is still same. Px can be always well-defined? Jun 4, 2018 at 8:35

1)

You are speaking of a new sample space $X(S)$, but usually not the image of $X$ is used for that but the codomain $\mathbb R$ of function $X$.

2)

The powerset of $\mathbb R$ is indeed a sigma field, but is not used in the new probability space. Practicized is probability space $(\mathbb R,\mathcal B,\sf P_X)$ where $\mathcal B$ denotes the sigma field of Borel sets. $X$ being a random variable means that $X^{-1}(B)$ is an element of sigma field $\sf F$ for every Borel set $B$, so $\sf P_X(B)$ is well defined for every $B\in\mathcal B$.

3)

If in your book $\sf P_X$ is named a "probability set function" then that is the naming of what I would call a "probability measure". I cannot imagine that they would mean "probability density function" which is another concept.

• I’m so sorry. I was confused. This books deals with the discrete random variable. And It doesn’t say Px is a probability density function! Sorry 😐 . It says small px is. I wrote it on the comment above. Thanks :) Jun 4, 2018 at 8:47
• The main thing is that you learned. You are very welcome. Jun 4, 2018 at 8:49
• Another thing. If it concerns discrete random variables then not what is called a PDF arises but a PMF. A PDF is defined as density wrt the Lebesgue measure, and a PMF (probability mass function) is a density wrt to a counting measure. Jun 4, 2018 at 8:54
• Thanks. And my book says a random variable is a function X which assigns to each element $c \in S$ one and only one real number x. This definition is wrong? Or your definition of a random variable is a different thing? Jun 4, 2018 at 8:55
• Oh..! Okay thanks! I think I need to study more 😢. I’m just studying an introduction for mathematical statistics. Jun 4, 2018 at 8:57