They are not the same thing, wrt most college texts. Range $\subseteq$ Codomain.

@amWhy I know. I used codomain because it's correct, but I recall most college textbooks not defining a codomain, but only a range. That's why I'm concerned people won't find this question. So changing it to "range" in the post wouldn't be quite correct, but might make it more easily find-able by these college students reading college texts.

I was confused by @amWhy's comment because I though codomain=range. I went down the rabbit hole and turns out "range" is ambiguous and people write it to mean either codomain or image (and I should have checked Wikipedia first, which mentions this ambiguity and says that the meaning changed from "codomain" to "image" over time).

During my school and college education, both codomain and range were defined and clearly differentiated for us. (This is in India, but even then my experience may not be universal in my country, so please take this only as one data point.) On the other hand, the body of your post includes the word "range" too, so your post ought to turn up under the relevant search parameters. Oh, and maybe I should add that I could be included under the category of "kids these days". (:

For example: $f: \mathbb N \to \mathbb N, f(n) = n^2$. The codomain is $\mathbb N$, but the range is $\{n^2: n\in \mathbb N\}$

@MikePierce: I don't quite understand your question. In real analysis, if a function $f$ is defined implicitly by an equation such as $f(x)=x^2$, then the domain of $f$ is assumed to be $\Bbb{R}$, and if so then the range (image) is $\Bbb{R}_{\ge0}$. But unless you specify explicitly what the codomain is, then it is ambiguous. It could be $\Bbb{R}$, or $\{z\in\Bbb{C}:\Re(z)\ge-\pi\}$, or it could be $\Bbb{H}$. So there's no way of "finding the codomain" unless it is explicitly specified. The codomain could be any superset of the range.

@Joe as a set everything in my post is the real numbers. But what are those real numbers? The domain of the PDF are peoples' heights expressed as a real number. The codomain of the corresponding CDF are proportions of people expressed as a real number. The codomain of the PDF is also real numbers, but what are they?

@TheAmplitwist I figured most of the responses here would be data points, or a discussion speculating at data points ;) As a data point myself, I don't think I heard the word "codomain" until I took a set theory class as a math major.

@MikePierce: So, you are saying that it is clear what the codomain is, but you are interested in knowing what the values in the codomain represent (in terms of people, heights, etc.)?

@Joe Yeah! Is that not clear from the body of my question?

@MikePierce: I think it is clear from the body, but your title misled me: when you said "what's the codomain", I read it as "how do you find the codomain?", not "what do the values in the codomain of a probability density function represent?" But I did read your question too hastily, sorry.

@amWhy: Do you mean $\Bbb{N}$ or $\Bbb{N}$? :)

@Joe No, I mean $\mathbb N$ $\mathbb N$ ;P

If you think there could be confusion, can you address it directly? Like "codomain (as distinct from the range", or vice versa. (Also, as single point of data, I'm a bit of an old, and was first taught the term "range of a function" in high school, for what is strictly the codomain. And what you are calling "range" was called "the image".)

Honestly, though, I think that question's title would best be improved by using the term "units" in it.

Perhaps the title of the question (on main) could be, How should we interpret the codomain of a probability density function?

@JonathanZsupportsMonicaC No? It's a bit of a waste of a reader's time to have them sort through the history of "range" vs "codomain" when they're just looking for an answer to question of how to think about that mathematical object, regardless of whether they call it a codomain or range. Really the underlying question is the same regardless of whether it's the codomain or range, regardless of a reader using the modern meaning of those words versus the old meaning. Like, maybe this meta post could've been more direct: How do we maximize the find-ability of this question for future users?

For the record, I'm a "kids these days" and had no idea there was a historical shift in the meaning of those words.

Well, I'm only one data point. Though I'd be curious if anyone else had the same experience.

For what it's worth, I don't recall codomain ever being much mentioned outside of category-theoretic stuff until maybe the mid 1980s, and even 20 years later the word was almost never used in (U.S. level) college algebra and precalculus and first year calculus. "Range" pretty much always meant the image set of the function, and the codomain of a function was hardly ever referred to (by name), although the notation $f:A \rightarrow B$ was of course commonly used. In rare cases where you needed to specifically mention the codomain, people sometimes incorporated the phrase "ambient space".