Awesome! Thanks so much for the positive note, and welcome!

This is wonderful to hear, @mehtubbhai9709! Really appreciate you taking the time to reach out!

Thanks, @braiandeivid! Appreciate the kind words.

Thanks, @omgupta2012! I've actually been slowly writing up my own explainers (yet to be posted) which will be my own version of a 'text'. A way to expand on some of the points not expanded on in the videos. I really enjoy 'Introduction to Statistical Learning'. In particular, the newest version now demonstrates how to apply the concepts in Python, which I think is fantastic. Conversely, I've always enjoyed Ang and Tang's 'Probability Concepts in Engineering' for a good overview of probability. It is particularly helpful, I find, for those who have some background in engineering to understand where these concepts may show up! Happy reading, and thanks again for the positive note!

@RiskByNumbers now I'm more curious about the text which you are writing, and the knowledge in that. Waiting eagerly for your explainers.❤️❤️

@washington_pc3306: thanks for the note! Your reasoning is very intuitive and sound. In this particular case, you are correct, which is due to the 'linearity of expectation'. If I sum multiple random variables, their expected value (i.e., mean) is the sum of their individual expected values. As the expected values for these 2 reports were 3 and 4 hours, respectively, then their summation is equal to 7 hours. Now, a couple of nuances. Linearity of expectations speaks to the mean. The mean may differ from the median (i.e., 50th percentile), which may complicate things. Second, determining the probability of exceeding, say, 9 hours is actually a bit involved. One distribution to check out is the Irwin-Hall distribution, which is the distribution for the sum of 'n' independent uniform distributions lying between 0 and 1. It is quite an interesting result. To solve this problem in the video, one can use convolutions. The nice aspect of Monte Carlo is that it can get you 'approximately' the right answer pretty quickly these days. And, furthermore, there is usually so much uncertainty in the underlying distributions (e.g., is Report 1 really going to take 1-5 hours, or perhaps 0.5-5.5 hours?) that an approximate answer is usually quite good. Thanks again for the comment. Feel free to follow up with me!

Hopefully higher than the probability that Todd causes you to be late to the party 😂

@tommasoc.2207: great to hear -- thanks!

Great example. The original plan for this video was providing the example of the value of a financial option, so it is great to hear that this concept resonated with your past experience!

@@RiskByNumbers In fact, ignoring transaction costs and fees, etc, the excess return on a 2x leveraged ETF expressed as a factor on the original investment is roughly the SQUARE of the excess return (also expressed as a factor) on the underlying reduced by multiplying by function of the total-variance (<1) due to slippage from “buying high / selling low”. For a 3x leveraged ETF, read “CUBE” and the total-variance reduction factor is lower, etc. This analysis can be shown to be exact in the limit of continuous trading from applying Ito’s lemma - which itself is a statement exemplifying the “flaw of averages” - and, for a given underlying return and total variance, the return on the ETF is NOT path-dependent, (but the margin is too small for me to put the proof here 😊) NOT investment advice or a recommendation to buy or sell any financial instrument. For academic interest only.

@jonmacdonald1413: thanks for this comment and for taking the time to visit the channel. Really appreciate it.

Great question, @TimschneiderSchneider! As highlighted at the start of this video, we are oftentimes interested in E[f(x)]. We want to know the distribution around some response, say 'Z', which is uncertain due to uncertainty for 'X'. In this video, we can directly determine the distribution for 'Z' (i.e., travel speed) by simply mapping the probability of each possible outcome for 'X' to 'Z'. However, that is not always the case. Or, it may be possible, but the problem we are looking at is quite large. Simulation is what I've found to be the quick solution in these cases. However, analytical approaches do exist, such as the use of Taylor expansions. Again, very good question(s).

@denizersoz7012: thanks for the clarifying question! Indeed, I found myself nervous using a speed example in a discussion around 'averages'. By 'average speed', I do mean total distance over time. By 'expected average speed', I mean the expected value across all possible average speeds. Here, we have three possible average speeds: 10, 20, and 30. As some have mentioned, this case is a great problem to highlight the harmonic mean. I'll think about how to fit that into future videos. Feel free to reach out to me directly to continue the conversation -- Omar

Thanks, @DenisZlokazov, for the comment! As I mentioned in a couple of responses, I was a bit hesitant around this example given the understanding of 'average' in a physics sense. Here, I do mean average speed as the total distance between 2 points, d2-d1, versus the total travel time, t2-t1. By its expected value, we can imagine three scenarios (given a deterministic distance): the travel time will be 120 minutes (10 mph), 60 minutes (20 mph), or 40 minutes (30 mph). With the travel times listed above, it becomes quite obvious what the expected travel time should be. Thanks for the comment and feedback.

@adelshahbakhsh2683, awesome and welcome!

@alex-craft: great comment and question! Having looked at log returns for financial assets in the past, I've found it to be a good 'approximation', but there is definitely evidence out there that they do not perfectly follow a Gaussian distribution (nor a random walk with drift). I think Black-Scholes is a nice example of a case where strong assumptions have been made that may not be perfectly true, but that does not mean that the Black-Scholes model has not been immensely valuable. A very interesting paper around the modeling of returns can be found in Andrew Lo's paper: "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test". It is a great, interesting read around the nature of stock prices explored through an intuitive variance ratio test. Highly recommend. Happy to continue the conversation, and thanks again for the question.

Thanks, @theondon, for the feedback! Great idea and food for thought.

Thanks, @vlad_objective! Very much appreciated.

I believe it comes from "The Undiscovered Self": "If, for instance, I determine the weight of each stone in a bed of pebbles and get an average weight of 145 grams, this tells me very little about the real nature of the pebbles. Anyone who thought, on the basis of these findings, that he could pick up a pebble of 145 grams at the first try would be in for a serious disappointment. Indeed, it might well happen that however long he searched he would not find a single pebble weighing exactly 145 grams."

@@RiskByNumbers was this written b4 or after the mathematics you referenced in your video??

@hdthor: this is a wonderful comment. A couple of others have also mentioned that the harmonic mean is something worth bringing up (I've discussed the geometric in the past), so I'll think about how to do so in future videos. Thanks again for the great comment and feedback -- definitely helpful food for thought!

Haha, @RandomBurfness! Perhaps next video I’ll try to throw in the word ‘datum’. Appreciate the note and subscription!

@lioncaptiv: wonderful to hear from you. Really appreciate the kind words, and thanks for supporting the channel!

@orterves: that's wonderful to hear -- much appreciated!

@deepakbhaiya.shorts: thank you so much for your kind words! Really appreciate it.

@Rdffuguihug: many thanks - cheers!

@ClearerThanMud: this is such an excellent example. Thanks for sharing!

I wouldn't be so quick to dismiss the idea! The proposed metric lies between the median and the actual average, which seems useful! The actual average might be of little interest when you know the total time. The median might be bad since it goes to 0 faster. It's a tradeoff between two location measures.

@MrHaggyy: thanks for the note -- you brought up some wonderful points.

@vitalysarmaev many thanks!

@andrashorvath2411 really appreciate the kind note and message! It means a lot. Cheers! -Omar

Thanks, @kevon21, for the kind note!

@leo_tra: great questions! First, in terms of applications, I'll note a couple of items. Jensen's inequality can show up quite a bit for certain proofs important in the areas of probability, statistics, and machine learning. A common example is Kullback-Leibler divergence (where we are measuring the difference between 2 distributions), where Jensen's inequality can be used to prove its non-negative property. For myself, the more interesting application of this idea is in the modeling of dynamic systems. A very intuitive example that I use to motivate one of my courses is the construction of a parking garage to maximize financial return. It costs you a certain amount of money to build each floor. Your revenue is based on demand for your parking facility, though there is a capacity constraint. You now want to balance your added revenue with each floor versus its cost to build. For an average demand, E[X], I can determine the design that maximizes my profit, f(E[X]). However, the expected profit, E[f(x)] is likely lower for that design when we introduce uncertainty. If demand is low, we are making less than we thought. If demand is high, the capacity constraint comes into play, and we can't take advantage of 'good times'. The solution then, in this uncertain system, may be to create a garage that is smaller (so that, if demand is low, we spent less upfront) but with beefed up columns and a foundation (so that, if demand is high, we can take advantage of that high demand). The reason that I like this example is that it does not require a background in optimization, reinforcement learning, etc. to appreciate that recognizing uncertainty allows you to identify adaptable policies that allow you to do well in the real world. As I worked out the script for the above example, I realized it was getting to be a bit much, which is why I just did this travel speed example. In terms of references, Warren Powell has provided excellent references on the topic, and he does a good job of trying to unify the perspectives of those working in the areas of optimization and reinforcement learning. Feel free to shoot me an email if you'd like to know more -- I'm planning to expand on the topic soon. Cheers -- Omar

@@RiskByNumbers Thank you for the details. I`ll try to go through Powell's work and if I have further questions I`ll message you.

@PeterZaitcev: thanks for the comment -- very much appreciated. I mentioned a couple of points in other comments that I'll mention here. Indeed, when I put together this video, I got a bit worried about using the term 'average' speed in the same video where I would be discussing expectations/means. Here, I meant by expected average speed as travel distance over time, as you noted, but considering that there are 3 possible situations. There is a 25% chance that total distance to total time is one ratio, etc. Now, the original problem was going to be a case of modeling a financial option, as I find them to be a great example of this idea. I actually will buy one each term in my class to motivate the topic and keep folks in the class engaged. They require knowing quite a few terms, though, hence I switched things up. Very much appreciate the feedback -- it is helpful as I think through ways to improve things in the future. Thanks again.

What is “true average”? Wouldn’t that literally be a mean of multiple speeds over time?

@@ryanlohbrunner7760 if you were driving for 1 hour with a speed of 20 km/h and last 2 seconds with a speed of 100 km/h, would you say that you average speed is (20 + 100)/2?

You're thinking of the average speed of a trip, which by definition would be the actual average speed of that specific trip. The video would make no sense at all if that's what he was talking about. He's talking about a trip with an unknown true average speed, but whose average speed is some distribution based on traffic that day, etc.

The speed given at the start of the video is not "average speed" it is "expected average speed". In fact, it was just written as "expected speed" omitting the words average entirely in the video graphics, because it's not the important bit. The important word is "expected". "Average speed" is just total average speed of a trip computed the natural way as you say. "Expected average speed" is the mean of those calculated average speeds over many trips. This also gives a useful intuition as to why they are different: "average speed" of a trip is a time-weighted average of your travel-speeds. "Expected average speed" is not a time-weighted average, it's just one data point per trip regardless of how long that trip took.

It is wonderful to see all of these great comments. Apologies as well for just chiming in -- we just got back from the hospital with our newborn, so a bit sleep deprived. @OMGclueless: spot on! I subtly mentioned 'expected' value at the start of the video to try and highlight a really important point. The motivation for this video (outside of trying to distill Jensen's inequality in an understandable manner) stemmed from some of my past consulting experiences I've done outside of my university day job. I have found that there is a tendency to frequently only discuss and use 'expected values' in making decisions without recognizing that, for non-linear systems, it may be quite important to know the underlying distribution for your random variable of interest. My hope is that this video helps clarify that point. The original motivation for this video was going to be determining the value of a financial option (e.g., call or put), but I realized pretty early on that it would make more sense to use an example familiar to most everyone. I might, though, come back to that in the future (ideally when I put together a couple of videos around reinforcement learning).

@maths.visualization: great to hear from you. I'll work on cleaning up the code on my end and eventually share it on GitHub. We are welcoming a new member to the family this week, so apologies ahead of time for the delay!

@33gbm: absolutely spot on and correct! One of the motivators for this video was that I have worked quite a bit with industry over the years and found that there is a tendency to avoid modeling uncertainties and to operate in a deterministic world. The consequences of doing so are not necessarily intuitive or apparent, particularly for more complex problems. Therefore, the goal was to show the possible consequences in a more straightforward example. Great job and catch! - Omar

​@@RiskByNumbersthanks for the reply. Nice to hear from you. By the way, the discussion about the problem is very well presented and I hope to see a lot more from you! 😊

Thanks for the positive comment, @boltez6507, and feedback. Really do appreciate it.

@theupson: love this point. I debated if it would be worthwhile to delve into that towards the end of the video, but the video was already feeling a bit long. Great point again, and I'll see if I can bring this up in a follow up video. Thanks!

Thanks for the comment, @fibbooo1123! Here, you could compute the inverse of the expected value of 1/X. E[1/X] = 1/4 x 1/10 + 1/2 x 1/20 + 1/4 x 1/30 = 7/120. Therefore, 1/E[1/X] = 120/7. We can then plug in our distance and 20/(120/7) = 140/120 = 7/6. Hopefully I've made no mistakes (welcoming a newborn to the family this week, so I've been a bit out of it...). Thanks again for the comment, and great catch around the relationship and importance of the harmonic mean!

@@RiskByNumbers looks right to me, thanks!

@@fibbooo1123Harmonic mean works for average, when sectional speeds are known for equal lengths. This is actually the case here, where "1/4,1/2,1/4" probabilities determine the distribution along the *path* not over time. Because it's not "1/4 of time the speed is 10" etc, it's "1/4 of the path the speed is 10". Sure the speed 20 is twice as likely, but that's a nice multiple - so just use it twice, basically the overall average speed is the harmonic mean of "10, 20, 20, 30"

Thanks, @wstaempfli!

Haha, cheers and thanks, @berlinisvictorious!

Thanks for catching this, @ReneKnuvers74rk! You know, I kept looking at this die thinking 'something looked off...' but could not put my finger on it. This is a good point that, in the future, I should be willing to spend a few bucks for a stock image and spend my time solely on the animations. Thanks again!

Thanks, @NicolasChanCSY! Just updated the subtitle -- appreciate the comment and positive feedback!

@lonjil: I have gone over geometric means in a past video, but I really should also find a point to bring up the harmonic mean (and their application). Thanks for the comment!

Thanks, @Koroistro for the feedback and for providing this nice explainer to others!

Yep, also the computed 'average' of velocity "20" is per unit length. Lower speeds yield longer time at that speed, greater speeds yield shorter time. Travel average is however an average over _time_ so such average speed is always lower - because time spent with different speeds is inversely proportional to the speed.. When probabilities are given per 'equal length', the _harmonic mean_ has to be used to determine travel average, not arithmetic mean which gives the expected value, let's see this example: as the probabilities stand, the quarter of _path_ is done with 10, half the path with 20 and quarter of the path with 30mph. Since 20 is exactly twice as likely, we simply use it twice. Average speed of "10,20,20,30" is then 4 / ( 1/10 + 2/20 + 1/30) which is 4*30/7 = 20*6/7 mph. For 20 miles the travel then takes 7/6 h Do check about the harmonic mean, it gives the average speed when sectional speeds are known for equal lengths. Eg going up the hill 10 and down the hill 30, the average isn't 20, it's 2 / ( 1/10 + 1/30) = 15 enjoy and good luck!

That intuition doesn't help, actually.

@martinsanchez-hw4fi: thanks for reaching out to me. One of the challenges I've found is that there are a lot of resources out there. This is both a blessing and a curse. I usually have to pull from many different texts/resources in the classes that I teach. Which areas in particular are you looking to learn more about? For the videos, it has been an evolution. My first few videos were largely powerpoint, now I primarily make all animations in Python. I typically make a video, learn through that process, and then make a few improvements for the next one. Over time, those improvements have added up (hopefully!).

@winchesterdown: this comment caused me to imagine myself successfully arguing against a future speeding ticket based on some obscure math/proof. On a more serious note: I would imagine that an average speed camera measures your 'average speed' by computing the ratio of distance over time for 2 different points. Therefore, if your instantaneous speed was not constant over that distance, then in theory your instantaneous speed at some point was higher than that average. I suppose that I wouldn't mention this fact in court...😂

@@RiskByNumbers haha. Yeah I thought about it a bit and came to the same conclusion as you.