Timeline for Why are $X \sim U(-1,1)$ and $Y=X^2$ dependent?
Current License: CC BY-SA 4.0
24 events
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| Feb 5, 2023 at 7:29 | comment | added | Glen_b | Uncorrelated is not the same as independent. One way to think about independence of two variables is "knowing the value of one variable tells you nothing about the distribution of the other that you didn't already know before specifying that value". When $Y=X^2$ that's not that case. For example, if you don't know $X$, then $Y$ is somewhere between $0$ and $1$ (but more often closer to $0$ than $1$). Now if you know $X=0$, $Y$ must be $0$, but if $X=1$, then $Y=1$ ... that's very strong dependence since knowing $X$ tells you everything about $Y$. | |
| Feb 5, 2023 at 2:08 | comment | added | kjetil b halvorsen♦ | @whuber: Yes, very uninteresting, but neither very difficult to find. | |
| Feb 5, 2023 at 0:00 | history | tweeted | twitter.com/StackStats/status/1622022212080340999 | ||
| Feb 4, 2023 at 21:33 | comment | added | Sextus Empiricus | If you think of stochastic functions, like the functions that map the distribution of $X_k$ to the distribution of $X_{k+1}$ describing stochastic processes such as auto-regressive processes or markov chains, then depending on the type of variable $X_k$ the distribution of $X_{k+1}$ can be independent of $X_{k}$, also for more interesting functions. (a situation described here has causal dependency, but not statistical dependency) | |
| Feb 4, 2023 at 20:55 | comment | added | Sextus Empiricus | For specific distributions more interesting functions can be possible. For instance, $Y = \sin(2\pi X)$ is independent from $X$ if the domain of $X$ is integer values only. But yes, it will still be the not so interesting map where all values in the domain of $X$ need to be mapped to a single point. It is because $f(X)$ given $X=x$ is a singular distribution in the point $f(x)$. Independence requires that the distribution of $f(X)$ is the same distribution for every value of $X$, $f(x)$ must be a single value for every $x$. | |
| Feb 4, 2023 at 19:39 | comment | added | Aksakal | it's strange question. if you observed the realization of X then you certainly know what is X^2. why would anyone even doubt they are dependent? they even have the same X in the definition of the variables. | |
| Feb 4, 2023 at 19:20 | answer | added | Sextus Empiricus | timeline score: 2 | |
| Feb 4, 2023 at 18:00 | comment | added | whuber♦ | @Kjetil Certainly. Not very interesting, are they? ;-) | |
| Feb 4, 2023 at 17:25 | comment | added | kjetil b halvorsen♦ | @whuber: one such function $h$ is $h(x)=0$. Another one is $h(x)=5$. | |
| Feb 4, 2023 at 15:55 | comment | added | whuber♦ | I find it challenging to find any function $h$ for which $X$ and $h(X)$ are independent. | |
| Feb 4, 2023 at 14:54 | history | edited | Richard Hardy | CC BY-SA 4.0 |
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| Feb 4, 2023 at 13:15 | answer | added | Massimo Ortolano | timeline score: 7 | |
| Feb 4, 2023 at 13:12 | answer | added | Henry | timeline score: 3 | |
| Feb 4, 2023 at 12:01 | history | became hot network question | |||
| Feb 4, 2023 at 9:05 | comment | added | Dave | @RaySiplao I would focus on the picture here. | |
| Feb 4, 2023 at 8:46 | vote | accept | Ray Siplao | ||
| Feb 4, 2023 at 5:26 | history | edited | User1865345 |
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| Feb 4, 2023 at 4:32 | answer | added | User1865345 | timeline score: 11 | |
| Feb 4, 2023 at 4:29 | answer | added | Zhanxiong | timeline score: 11 | |
| Feb 4, 2023 at 4:15 | comment | added | Galen | Plotting simulated values of $X$ and $Y$ will be enlightening, but you can also work out the density of $Y$ via a change of variables from $X$ in order to compare the probabilities in terms of definition of statistical independence. | |
| Feb 4, 2023 at 4:08 | comment | added | Ray Siplao | @Dave RVs are independent if $P(X\le a, Y\le b)=P(X\le a) P(Y\le b)$ for any real number $a,b$. I am asking this question because I don't understand some answers in this post. | |
| Feb 4, 2023 at 4:02 | comment | added | Dave | Welcome to Cross Validated! What’s the definition of independence? | |
| S Feb 4, 2023 at 3:55 | review | First questions | |||
| Feb 4, 2023 at 4:12 | |||||
| S Feb 4, 2023 at 3:55 | history | asked | Ray Siplao | CC BY-SA 4.0 |