Timeline for Why do we need to define a random variable as a function?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27 at 20:50 | answer | added | Michael Hardy | timeline score: 1 | |
| Jun 27 at 20:27 | comment | added | Christian Remling | A pdf can not be "used to define a function." It is true (and I suspect this is the source of your discontent) that one often reads things like: "let $X$ be a rv with $P(X=0)=1$; find $EX$." This does not mean that we were just given the definition of $X$. Rather, we were only given partial information (the distribution of $X$, rather than the full function together with the prob measure on the sample space), and this usually sufficient to answer probabilistic questions about $X$. The practice is lamentable in a pedagogical setting but widespread. | |
| Jun 27 at 18:25 | comment | added | LSpice | This seems like a near duplicate of Why do we need random variables? | |
| Jun 27 at 18:24 | history | edited | LSpice | CC BY-SA 4.0 |
$\sigma-$ -> $\sigma$-
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| Jun 27 at 18:22 | comment | added | Jannik Pitt | @QiaochuYuan You push-forward by $f$, which needs to be measurable – which is exactly the definition of a random variable! | |
| Jun 27 at 18:13 | comment | added | Qiaochu Yuan | @Jannik: you can do that using measures, just take the pushforward along $f$. | |
| Jun 27 at 18:03 | answer | added | Qiaochu Yuan | timeline score: 4 | |
| Jun 27 at 17:27 | review | Close votes | |||
| Jul 5 at 3:03 | |||||
| Jun 27 at 17:26 | comment | added | Jannik Pitt | @Syail Sure, one could develop everything just using measures without really mentioning random variables. But what random variables allow you to do conceptually, is to have multiple random phenomena on the same sample space. I.e. it gives you a conceptual framework of e.g. asking "Given a length $X$ I measured, what is the probability distribution of $f(X)$", where $f$ is some function taking measured lengths to something else. | |
| Jun 27 at 17:23 | answer | added | Iosif Pinelis | timeline score: 3 | |
| Jun 27 at 17:17 | comment | added | Syail | If we define a random variable to be a probability measure $P$ on the Borel sets, then the probability $X$ in $S$ is exactly $P(S)$, if I am not mistaken. | |
| Jun 27 at 17:09 | comment | added | მამუკა ჯიბლაძე | What would be for you a more intuitive definition that would allow you to answer questions of type "what is the probability that the value of $X$ is in $S$" for a measurable set $S$ of real numbers? | |
| S Jun 27 at 17:00 | review | First questions | |||
| Jun 27 at 17:09 | |||||
| S Jun 27 at 17:00 | history | asked | Syail | CC BY-SA 4.0 |