3 ms·
If randomness is evenly distributed in a random walk, it tends to stay close to the starting point. Edit: Plot of steps Timestamp: 34:49 https://ocw.mit.edu/c
by Attummm 3y ago
If randomness is evenly distributed in a random walk, it tends to stay close to the starting point.
Edit:
Plot of steps
Timestamp: 34:49
https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/resources/lecture-5-random-walks/ https://ocw.mit.edu/courses/6-0002-introduction-to-computati...
- 082349872349872 3y agoIt'll gradually creep away*, proportionally to sqrt(t). Got a timestamp in that video that says differently? * in the absence of (restoring) forces, it's unwise to rely on things staying where you left them, as that'd be betting against entropy. Lagniappe: https://www.youtube.com/watch?v=i6rVHr6OwjI https://www.youtube.com/watch?v=i6rVHr6OwjI EDIT: even more apropos: https://www.youtube.com/watch?v=ObvxPSQNMGc https://www.youtube.com/watch?v=ObvxPSQNMGc
- Attummm 3y agoThink of it this way: if randomness is evenly distributed, the steps to the left and right will balance out, just as the steps up and down will. This balance tends to keep the random walk close to its starting point.
- 082349872349872 3y agoExcept they don't balance. (if they do balance, your walk is not very random. https://en.wikipedia.org/wiki/Cyril_Burt#Scientific_misconduct https://en.wikipedia.org/wiki/Cyril_Burt#Scientific_miscondu... was caught faking data because his residuals were improbably small) Exercise: how many strings of length N with the alphabet {R,L} are there where #R = #L? Exercise: as above, but for #R = #L + k See also Haldane, "The Faking of Genetical Results" pp8-10 in https://archim.org.uk/eureka/archive/Eureka-6.pdf https://archim.org.uk/eureka/archive/Eureka-6.pdf
- kqr 3y agoThey can balance but still be random -- that's the essence of drawing without replacement, or the hypergeometric distribution. What you probably mean is that they are no longer independent.
- 082349872349872 3y agoYou're right. (I'd count enforcement of "without replacement" as a force; at least for classical walks it'd require some kind of Maxwell daemon?) I guess what I want to get at: a Galton board is easily understood by elementary school students (and probably constructible by those in the upper grades), and it doesn't take much time playing with one to notice that all the balls don't wind up with their paths balanced out, in the middle column. (even the normal distribution is misleading, because centrality is a 1D phenomenon; given enough dimensions observing the exact average becomes exceedingly unlikely)
- kqr 3y agoIf you take the same Galton board and focud on the paths taken by all balls that ended up in a particular bin you get the hypergeometric distribution for the steps they have taken. So the draws without replacement lurk in the Galton board too. It does, in other words, not require a demon, only a restricted viewing angle.
- 082349872349872 3y agoTIL! > Page unintentionally left blank. —CS Skämtar du med mig?
- rcxdude 3y agoAnother way to think about it: the more steps, the more likely it is you get a run which is imbalanced for long enough to make it some given distance from the origin.
- Sharlin 3y agoThat balance will only tend to keep the average coordinate near the starting point. But the variance/std dev will increase without bound.
- Attummm 3y agoAlthough you are correct, the goal of the comment was to illustrate why random walks would gravitate towards their starting point.