What is a Random Walk? | Infinite Series

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Markov Chains

Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux

Random Walks are used in finance, computer science, psychology, biology and dozens of other scientific fields. They’re one of the most frequently used mathematical processes. So exactly what are Random Walks and how do they work?

Comments answered by Kelsey:

Petros Adamopoulos

Jonathan Castello

Niosus
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And that is why so many biological reactions take place on membranes. Instead of molecules bumping into one another in three dimensions (free in solute), they are confined to two dimensions (on a membrane). Speeds up the processes greatly.

Valdagast
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First, we need to establish a ministry of random walks.

Scorpionwacom
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11:50 YES! there is an amazingly beautiful algorithm for finding the area of ANY closed non-intersecting polytope using recursion. So simple that if you are familiar with the divergence theorem (which obviously you are but others might not be), I can just throw the proof in a couple(ish) lines in this comment.

The "volume" V (content) of a polytope can be written as the integral:
V = ∫1dv (where we are integrating over the polytope)
However, since the divergence of 𝐱 is n, 1 can be rewritten as (1/n)∇∙𝐱 where 𝐱 is the position vector and n is the number of dimensions of the polytope, so V becomes
V = (1/n)∫∇∙𝐱dv
but we can apply the divergence theorem here and get
V = (1/n)∮𝐱∙dA (where we are integrating over the "faces" (facets) of the polytope)
but now we can split this integral into a sum of integrals over each face
V = (1/n)∑∫𝐱∙dA
which simplifies to
V = (1/n)∑∫𝐱∙𝐧dA (where 𝐧 is the normal of the facet)
but since each facet of the polytope is flat, by definition 𝐱∙𝐧 is constant.
so pick any point 𝐱₀ on the facet and this turns into
V = (1/n)∑𝐱₀∙𝐧∫1dA
but ∫1dv is just the area A of the facet, so

V = (1/n)∑𝐱₀∙𝐧A

So this means that to find the content of any polytope, all we need to do is sum the areas times any point dot the normals of each face and divide by the number of dimensions of the shape, (where the areas can each be found by this exact same process recursing down until we get to edges or vertices (where vertices have content 1)) and then the we have the content.

Now maybe it's just me, but I find this result to be extremely cool in the sense that it means that you can write a function of only about 5 lines that will be able to compute the area, volume, hypervolume, or whatever else of any non-intersecting polytope in a very flexible and general way.

Now, to address the confused computer scientists out there, I do realize that this formula requires the normals of each facet, and that in order to get each normal, you need to take a square root to get the length of a vector which is a computationally expensive task. However, I'd still argue that this is still a cool algorithm. Also, there may be a way to avoid square roots by cleverly using squared lengths and certain ways of computing normal vectors, but I'm not sure.

Also, I do still believe that this is technically an O(n) algorithm. I am not a computer scientist or student, so I am not sure, but considering that through recursion this effectively just boils down to a sum over vertices, I do think this would be considered O(n).

(and finally, I do realize that I totally just pulled a math and said something was "amazingly simple" and subsequently used things like multivariable integrals, vector algebra, and the divergence theorem which I don't think everyone here is familiar with, but I hope this still was informative for some people (and that I didn't make some glaring mistake))

haniyasu
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This channel is seriously making so many of the concepts underlying the stats I'm learning in grad school intuitive! Such a great video.

ProfessorPolitics
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1:10 - 1:55 Is that a Pascal triangle? I'm pretty sure it is. But why?

EDIT: I was being silly. *Of course* it's a Pascal triangle. After all, we made this by the same process - by adding the values of the two neighbors on the one higher level. Disregard my question, I didn't think.

gressorialNanites
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i find your description of math as both a practical and intellectual property fascinating. please keep it up.

tristandoerksen
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I like this girl! She's so smart!

Bibibosh
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Hi! I love this channel very much and I'd like to make a suggestion.
I really enjoy understanding the logic behind an equation but when you provide us with a simplified answer such as -sqrtN or sqrtN I have no idea what the equation looked like before simplification unless I grab a pen and pencil, which is often times inconvenient. Do you think you could always show the expanded or semi expanded equation before showing the simplified one? It'd really help a lot!

doodelay
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my favorite random walk are the photons trying to scape from the sun and taking hundreds of years to do so

falnica
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I do not get the intuition of a random walk being transient at above 2D. Yes, the walk has more wiggle room in 3D, but so does a 2D compared to 1D.

konstaConstant
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Thanks for this awesome video! It was just the perfect amount of depth to gain an intuitive understanding of Random Walk, and to spark a interest about this topic to actually start reading the textbook of my CS probability course! Subscribed!

JasonDangol
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2 steps forward, one step back; you're still moving forward.

darkracer
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My favourite use of Markov chains was my own use of them in my MSc Mathematics dissertation. I used Markov chains to represent disease transmission - my states were what stage of the disease you can be in. I compared their effectiveness to standard SIR black box mathematical models and they were surprisingly accurate!

mattfeeder
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Thank you for the excellent description of the theory, especially for the pleasant appearance and voice of the narrator and the beautiful video design. 👌

ashotamyan
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Love the topic of randomness, thanks for this. Seeing the graphs reminded me of electron clouds and excited fields, and it made me wonder; do random walks actually apply in those scenarios?

AltisiaK
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Congratulations on this video and the channel !!! I really enjoyed this information. Thank you.

EdsonVazLopes
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another application for people interested in fluid dynamics: random walks are one way to model how small droplets or particles are tossed around by turbulent fluctuations.. Quite important for all sorts of environmental and industrial flows.

mortenmeyer
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you are way better than my prof in explaining this, i dont know how my prof was even hired to teach this course

haoyuansun
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I have also noticed that the 1d non biased integer walk after n steps looks like the nth step of the summing triangle: 1, 1(/2); 1, 2, 1(/4); 1, 3, 3, 1(/8); 1, 4, 6, 4, 1(/16 and so on). But the random walk resembles a bell graph. So the summing triangle as some mysterious connection with the bell graph.

pietrocelano
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Random Walk @The thoughts you think are magnetic: if you think positive=u attract positive things, if you think negative=u attract negative energies. If you think randomly (Random Walk) with more tendency toward positive= you move toward positive and vice versa. if ...

eimaldorani