Timeline for What is this distributional closeness?
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| Jul 9, 2023 at 11:39 | comment | added | Iosif Pinelis | Previous comment continued: Also, I don't think that during my 5 years as a university student (about 50 years ago) any one of our professors ever used "function $f(x)$" (again, unless $f(x)$ did happen to be a function). So, I must highly doubt your $99\%+$ contention (and also other contentions in your comments on this page). | |
| Jul 9, 2023 at 11:38 | comment | added | Iosif Pinelis | @mathworker21 : The first book I have opened since our conversation is Rockafellar's Convex Analysis (Section 13) , and it uses notations like "function $f(\cdot)$" rather than "function $f(x)$". I would be greatly surprised if Bourbaki ever said "function $f(x)$" (unless $f(x)$ did happen to be a function -- such as the linear functional that is the Fréchet derivative $f'(x)$ of a nonlinear functional $f$ at a point $x$). | |
| Jul 9, 2023 at 2:05 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 7, 2023 at 20:47 | comment | added | Iosif Pinelis | @mathworker21 : $\sin x$ can also be thought of as an expression, and then $\frac d{dx}\,\sin x$ can be thought as an expression, whose value at $x$ is the derivative of the function $\sin$ at $x$. More generally, $f(x)$ can be thought of as an expression. However, "function $f(x)$" is syntactically wrong, even if most people understand this phrase and are OK with it. | |
| Jul 7, 2023 at 20:40 | comment | added | mathworker21 | "the derivative of the sine function at $x$". but $x$ appears in the denominator $\frac{d}{dx}$. And, according to you, $\sin(x)$ is a number not a function. | |
| Jul 7, 2023 at 20:39 | comment | added | Iosif Pinelis | @mathworker21 : As for the post in question, my biggest problem with it is that it is not stated there that the distributions there are discrete (or maybe not?). People may and do write all kinds of things, including, as I said, things like "probability distribution $f(x)=e^{-x}$ for $x\ge0$" -- or sometimes even without " for $x\ge0$". So, I was unsure which kind of case I was dealing with. | |
| Jul 7, 2023 at 20:28 | comment | added | Iosif Pinelis | @mathworker21 : I do not see a big problem with your latter expression, $\frac d{dx}[\sin(x)]$. I think such an expression would be OK in an appropriate context. It would be bad, though, is someone said that $\frac d{dx}[\sin(x)]$ is the derivative of the sine function (rather than the derivative of the sine function at $x$). In any case, I would drop the brackets and the parentheses as quite unnecessary, to get $\frac d{dx}\sin x$. Also, I certainly hope that it is not $99\%+$ of all people who would write "function $f(x)$"; however, I agree that most people will understand this correctly. | |
| Jul 7, 2023 at 19:58 | comment | added | mathworker21 | @IosifPinelis It seems we just have a factual disagreement. I think $99\%+$ of people would write "function $f(x) = \sin(x)$" instead of "function $f = \sin$", and $99\%++$ would understand what "function $f(x) = \sin(x)$" would mean. My point is that I think most people would understand this OP's question, since his notation is very much in line with the norm. With that said I do agree with you in general (I always disliked $\frac{d}{dx}[\sin(x)]$, for example). | |
| Jul 7, 2023 at 17:04 | comment | added | Iosif Pinelis | Previous comment continued: Again in this case, I had an unpleasant choice to either change notations or use notations I am really uncomfortable with. I also had to assume the discreteness of the distribution. I think the OPs should try to spare readers of their questions of things like that, especially when it is so easy to do. | |
| Jul 7, 2023 at 17:03 | comment | added | Iosif Pinelis | @mathworker21 : Yes, I would object to someone saying "function $f(x)=\sin(x)$", especially when one can simply say $f=\sin$ instead. In this case, it seems to have turned out that the OP was using notation "informally", but I have encountered many cases when the OP indeed did not know what a distribution is and thought that a formula like $f(x)=e^{-x}$ for $x\ge0$ is a probability distribution. | |
| Jul 7, 2023 at 16:41 | comment | added | mathworker21 | @IosifPinelis I really disagree with you and think you are being a bit rude. It is very, very common to write $P(x)$ instead of $P(\{x\})$ for discrete probability distributions $P$. Also, writing $P_S(x) = P(x)/P(S)$ is analogous to writing $f(x) = \sin(x)$; you wouldn't object to someone saying "let $f$ denote the function $f(x) = \sin(x)$" by saying "$f(x)$ is a number, not a function". | |
| Jul 7, 2023 at 16:25 | comment | added | Iosif Pinelis | Then I think you should warn people about informal notations. | |
| Jul 7, 2023 at 16:24 | comment | added | user43170 | thanks, I agree that my notations are a bit informal. | |
| Jul 7, 2023 at 16:23 | vote | accept | user43170 | ||
| Jul 7, 2023 at 16:12 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 7, 2023 at 16:03 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |