Mathematics of Maximizing Profit in Gambling/Investing - Kelly Criterion

In this video, we introduce the Kelly criterion which is the formula that gives optimal risk that maximizes the long term profit, and we will proceed to derive the formula in a nonstandard way.
The necessary prerequisite materials like random variables, transformations of random variables, expected value, generalized mean are introduced in the video.

Links:
Proof that E(B) = np
Wiki page which has detailed information about the median and the mode of the binomial distribution
geometric mean is the limit of the power mean as the power goes to 0

Chapters:
00:00 Intro
03:18 Problem Statement
04:51 The Kelly Criterion
06:36 What is an Average?
11:53 First Step towards the Model
13:30 Random Variables
16:00 Expected Value
18:05 Transformation of Random Variables
20:20 Random Variable for the Problem
22:26 Median of Random Variables
29:15 Mode of Random Variables
31:18 Geometric Mean of Random Variables

Music🎵:

Corrections:
18:22 Y=1/X should be 1/X if X ≠ 0 and 0 if X = 0
21:59 E(R) = (1+r)^n, not (1+r)^20

[applimu]

20:52 Here is what the computer did!

First you start out with the sum from k=0 to n of (1 + tr)^k * (1 - sr)^(n - k) * Binomial(n, k) * p^k * q^(n - k)

and then you can combine some terms because a^k * b^k = (ab)^k, getting

the sum from k=0 to n of (p + ptr)^k * (q - qsr)^(n - k) * Binomial(n, k)

Then, due to the binomial theorem, this is simply equal to ( (p + ptr) + (q - qsr) )^n

And with a little simplifying you finally get (1 + ptr - qsr)^n :D

[andrewrhowe]

I am thoroughly impressed and grateful for the quality of your videos. Recently, I have been voraciously studying probability theory, statistics, and machine learning. Your videos explain things so well that I had to say something. By the way, I did not want to use YouTube's 'Thanks' feature-- I would much have rather subscribed to a Patreon or donated to you directly, but this was the only option, as I looked for other ways to support you! Thank you so much for your videos. They have completely taken priority over some of my lecture notes, the next chapter of Murphy's PML book that I'm on, and several videos by other creators. A few key videos of yours are next on my list for rigorous study thanks to the incredible bandwidth of knowledge transfer you provide. I also love your presentation style, with occasional humor and very well-placed context for certain problem solving decisions.

[RahilJain-xj]

When I clicked, I expected to learn something about investing. But only 11 minutes into this video, you've covered- Maths, Statistics and Chemistry

[christophesiewecke]

This video does a really good job of:
- explaining the algebra behind concepts like mean, median, mode
-showing math equations and formulae and what their terms represent
-going in depth on the math behind statistical and probabilistic principles

The editing is solid, although as people have pointed out, the audio recording on your mic has two different settings which is a bit jarring. The animation and overall visuals are excellent.

This video does a very poor job of:
-explaining basic algebra and calculus in a way that is comprehensible to people not well versed in math or who are just very much out of practice (like me!)
-giving a conclusion or interpretation to all of the concepts presented.

You can't leave the interpretation of a mathematical equation up to the viewer. This isn't a stats class in university. This is infotainment! I'm now going to go watch how the Kelly Criterion is interpreted elsewhere on YouTube so I can actually draw conclusions on it and base my investing behaviour on those conclusions. I understand your hesitancy to do so yourself as you don't want people to make bad choices based on your advice, but I still feel like you could have done more to explain what the criterion implies without giving investing advice.

I really appreciate the effort you took to make this video and as I said, there are many positive points, I hope you can appreciate my feedback as constructive, as no offense was meant. Maybe I'm not the target audience, but I strongly feel like an interpretive conclusion was missing in this video.

[davidawakim]

Great video! Thank you so much that was great at elucidating so many scary stats and probability concepts in one nice, clean mathematical umbrella! Did a better job in 30 minutes than half a semester of courses

[breadbmeat]

It was really cool to see that there were four different types of averages.

[lioncaptive]

This video gave me a lot to think about including the intuition of putting my money under my mattress.

[yudistiraashadi]

I'm dumb with math, and I English isn't even my first language, yet your explanation is so clear I can understand all of them. Amazing job!

[jonnyjazzz]

Very well explained. I use this concept to allow my trading bots to adjust Risk by thinking of the history of wins/losses as the changing probability, so it doesn't bankrupt me if market conditions become unfavourable or if there's faulty logic that I didn't perceive.

[gabrielcazaubieilh]

Very nice video, thank you! I think the proof of equivalence with maximizing the geometric mean can be much simplified by expressing the random variable ln(G) as a function of the ransom variable k, the number of wins

[Jacob.Cornejo]

I found a counterexample for the continuous case:

let X be a standard normal distribution N(0, 1) which has one mode at X = 0

then Y = ∛X is a transformed random variable, and the real cuberoot is monotonic,

but the probability density function of Y is
f(x) = (3/√(2π)) Exp[-x^6/2] x^2 which has 2 different modes

[jacobmears]

There are too few views on this excellent video

[bthomas]

math would be so much more understandable/accessible if mathematicians just used whole words/phrases to define variables instead of single letters. Like why do i need i hold in my head "r = risk" can you just say "risk"? what do you gain by abbreviating every little detail?

i'm sorry if i sound annoyed or obnoxious but for me it's so much more easier to grasp the full concept of what is being explained when i don't have to hold all these little bits of translation in my head at the same time. I'm sure there's some level efficiency gained by growing the mental muscles of holding multiple definitions in ones brain at a time. but there is such a thing as over abstracting variables to be placeholders that could mean anything.

this video seems really cool but i honestly just don't have the capacity to retain all these definitions just to understand what "p(w) * r / t - s" means. maybe my brain just works different or i'm dumb, idk maybe i'm just a software engineer who is so used to thoughtfully named variables that this level of abstraction just feel esoteric.

[pedrocolangelo]

Sir, these were the best 30 minutes I spent in YouTube all of my life. Thank you so much!!

[diarya]

If you have one bad bet, your leverage is gone and you're immediately in debt and in a bad place. On average you'd be fine, but if you bet *everything*...

[obelarcapital]

Bro the insurance company stealing money from you joke got me. I subscribed. Good work.

[alan.nartikoev]

18:41 The range of Y is actually given by {-1, Inf, 1, 1/2}.

Thanks for the great video!

[hogopipo]

Didn't realize I was so uneducated til I watched this video.

[TallestFiddle]

Thanks for the video, I enjoyed all the examples and metaphors in the beginning. I think it would be great if you had more in the second half. And especially to wrap up the video, some examples would have been great to drive the point home.

[icecream]

So i really disliked statistics class. This video, while i still have some dislike towards statistics, I want to relearn all of it again. Seems that statistics, out of all the branch of mathematics, is the most applicable in day to day life. Math class should teach more of the different types of mean, and how they can be used to solve questions from elementary school level to industrial real world stuff. It's not like it's the best, just another way of thinking the same issue and i find it really intriuguing

thanks for the video :)