A Simple Solution for Really Hard Problems: Monte Carlo Simulation 

🤣 @RichardRietdijk: one of the best comments received yet -- it made my night!

Thanks, @vitor00! Also glad that you were able to work out the analytical solution -- well done!

Thank you for the positive feedback and kind note -- I sincerely appreciate it!

Poor Todd, @zp5808. He's really just upset that he was not invited to the BBQ. 😂

@@RiskByNumbers 🤣

Definitely true! Thanks for providing this feedback.

LOL @jameslinzmeier368, touché!

Thanks, @matholive6994! Definitely a simple example that you could solve through alternative means. When I find the time, I'm going to create a blog where I can discuss in greater depth the different ways to solve a problem like this one. Thanks again for the kind words!

Thank you, @eamonmolloy8308! Sincerely appreciative of the kind note.

Thank you, @adamjc86, for the kind comment! Really appreciate it.

Thank you, @pedrocolangelo5844, for the really kind message! I'll be getting the next video finally out next week!

Thank you, @KpxUrz574! I really appreciate the positive feedback.

Thanks, @altr0403! Your kind note means a lot. Thanks as well for subscribing!

Thanks, @imyashbhatt! I appreciate the kind words.

Many thanks!

Thank you for the kind note! I really appreciate it.

This video is just crap. Nothing else!! Monte Carlo is NOT by any means used in statistics or data sceince.

Thanks, @othmanmurad5267! That was the goal of the video, so I'm glad it resonated with you. And, yes, definitely intending to put together a video on Markov chain Monte Carlo in the not too distant future.

Thanks for the kind words, @YuXingSong!

Thanks, @OutOfRangeDE, for highlighting another application for Monte Carlo! It has been neat to hear from many folks around how they've applied Monte Carlo simulation. Your comment sent me down a 30-minute rabbit hole of reading up on its use for circuit designs -- its been great to learn via the comments.

Haha that is awesome, @KinomaroMakhosini!

@bernardogeocometto5562: thank you!

@LiamRohan_: really great question, and thanks for the positive feedback! In terms of learning about probability theory itself, I find that there are less 'accessible' resources than those related to statistics. This may be due to the fact that it is quite easy these days to import different libraries and use them to estimate a model. Having said that, I think that MIT OCW has a great primer on probability via 18.05. If you want to purchase a text, Ang and Tang have a good text titled 'Probability Concepts in Engineering' that I've seen used for courses in civil engineering, nuclear engineering, mining, and others. Now, there are even less resources in terms of practical application of risk analysis. Richard de Neufville is one author/researcher who comes to mind that provides practical resources around risk analysis and modeling for real problems. He focuses quite a bit on flexibility (which, in principle, underlies methodologies such as reinforcement learning) to better handle uncertainty and discusses so via quantitative as well as qualitative means. Is there a particular application you are working on these days?

Thanks for the suggestion, @johnmaskell5124! I really enjoy the intuition and simplicity of Cholesky decomposition when introducing folks to generating correlated random variables, so I'll plan on adding that to the list of future videos. Given the time it takes to create each of these animated videos, I am thinking that I'll make a blog soon, where I can provide more rapid thoughts on these topics. I'll announce that once I've set it up. Thanks again for your comment -- Omar

🤣

Really appreciate the comment and feedback, @muzahmad2104! Now that I've received quite a few comments around distribution selection, I'm switching up the order of upcoming videos and having the next one focus on that topic. The comments received have been really valuable in determining what to focus on next. Thanks again for the feedback and positive note!

its a pleasure, look forward to it. Keep up the good work

Thanks, @briancruz3551! I appreciate the kind words. For the visuals in this video, I relied on a few tools. First, for the graphical animations (e.g., the Monte Carlo sampling), I wrote up a small Python script. I am doing that more and more in my videos, primarily with matplotlib.animation. As I use matplotlib so frequently for work, I have used to date, but this summer I'll explore other libraries. Second, for the screen recording I used OBS Studio (which is free!). Finally, to bring all the animated clips together, I use Premiere Pro. Launching this channel has given me a much greater appreciation around other channels that I watch on RU-vid in terms of their time and effort! Final exams end this Friday for me, so I'll be much more timely around my output soon!

Thank you! Trying my best to make sure next week's video is straightforward, too!

Thanks, @charliehunter_! It was meant to be the probability the deadline would be missed. I summed up the number of instances that exceeded a threshold (in this case 9 hours) versus the total number of simulations. So, 12.5% is approximately the probability it takes us more than 9 hours (implying the deadline is not met). Thanks again!

@Tommybotham: thank you!

@@RiskByNumbers I have a complex physics problem that I soon need to simulate, involving ionic particles (with a velocity distribution) that come in at different incident angles with some probability to neutralize on a surface AFTER which I'd then have to simulate the directions they go afterward. Monte Carlo came up and so I needed to brush up my knowledge and build some intituition to solve this problem. Your video greatly helped me get down some ideas on how to solve my problem :)

​@@Tommybotham thanks for the reply, and apologies for just getting back to you. I've really enjoyed learning about the different problems that others have analyzed/solved via Monte Carlo simulation. Glad the video could be of some utility, and I hope that you were able to solve it!

Thank you!

Thanks, @ambrosianas7505, for subscribing! I'll definitely continue to post content -- thanks again!

@kimsourtat601: great question! In this case, the expected value (the median happens to be the mean) of our two random variables is 3 and 4. The expected value of their summation is simply 3 + 4 = 7. However, ignoring the full distribution means that we cannot comment on other statistics (e.g., variance) nor explore, for example, the probability the total duration exceeds some threshold (in this case, 9 hours). Monte Carlo simulation, however, helps us do that without working out the solution analytically! I hope that helps.

@@RiskByNumbers Thank you so much, the expected value is 7 hours (my bad is 8 hours). In this case he has 9 hours left to complete the task but there is 12.5% chance he could complete, if he take this task, he is unlikely to join his party. : (

@a1cswiz1611: great question (and apologies for the delayed response!). For each of these two reports, we are assuming that there is uncertainty around to complete them. For Report A, it is 1-5 hours and, for Report B, 2-6 hours. We can use our probability density function to define this uncertainty. You could also use a probability mass function, if we wanted to model time to complete each report as a discrete random variable. In NumPy, you can do this via np.random.choice(), and you can alter the probability masses as you see fit (i.e., each outcome does not need to have an equal probability of occurrence). The key aspect here is that the probability densities/masses should be representative of the uncertainty. If you believe certain times to completion have a higher relative likelihood, you should select a probability density/mass function representative of that situation. The nice aspect of Monte Carlo simulation is that it can adapt to alternative distributions. I hope that makes sense. Do not hesitate to follow up if you have any further questions. Omar

I was planning on covering Markov decision processes eventually, so this would be a great moment to discuss these topics. I sincerely appreciate the interest in the channel, the suggestion, and the patience in awaiting new content!

Thank you, @danielzuzevich4161! Much appreciated.

Thanks! That is definitely something I can introduce. The next few videos will focus on a few algorithms (e.g., dynamic programming, reinforcement learning), but the content afterwards is still up in the air. Thanks again for the comment and request.

@taylorolp8090: really great question! The length of all 3 arrays is actually 1 million. As you mentioned, for "A" and "B" we have generated 1 million random values. When we implement "duration = A + B", we will actually create another array that is the same length as "A" and "B". The value for the first indexed value, duration[0], will be A[0] + B[0]. In fact, for any index value, i, duration[i] = A[i] + B[i]. In other words, we will have 1 million values, where each value is the sum of a random realization of A and another random realization of B. I hope that makes sense. Feel free to reply or follow up with me directly if anything is still ambiguous.

Thanks!

Awesome @raymondnepomuceno8815! Welcome to the channel! Really appreciate the support.

@dalrax4: really great set of questions! Thanks as well for reaching out! For our Monte Carlo simulation, we could have actually assumed our distributions came from any available distribution to us. In this video, I used NumPy, and as you can see we actually have a broad set of distributions available to us for random sampling: numpy.org/doc/stable/reference/random/legacy.html. Your point highlights an important aspect of Monte Carlo simulation. Unless we specify our distributions (reasonably) correctly, we may be facing a case of creating a model that is 'garbage in, garbage out'. This point also highlights why, in many cases, an individual may be okay with just 1,000 Monte Carlo simulations. Given that we typically do not know our underlying distribution (it is estimated through various statistical means) for our inputs, we are oftentimes okay with an approximate estimate of the distribution for our response variable of interest (in this case, the total time to complete both reports). Your point about stock market returns is great (and relevant to the channel)! I'll actually be talking about the usefulness of natural logarithms for modeling financial data in the next video. I've recorded the audio and have made the animations -- I'm just finalizing the stitching of the two together.

@@RiskByNumbersThank you for clarifying that!

@MrMeltdown: it's been pretty neat for me to see the range of responses to this video. Although the intended focus of this video was less so around the correct course of action for this situation, I enjoyed reading your response and your perspective. Personally, I have found that providing a rationale and acknowledging the other's perspective can go a long way in these situations. I find this even when I'm teaching. Explaining 'why' I've structured an assignment in a certain manner goes a long way in terms of student buy in. Similarly, acknowledging how student feedback from past years has informed certain choices in the current year gives students a sense of voice and acknowledgement. I think this extends to more traditional employer-employee relationships, too. Now, whether that actually happens is another story...

Thanks, @davidclercrodriguez5801! Really appreciate the kind words.

@statistik4908 you may be interested (if you have not read it already) in McKay et al.'s 1979 paper that compares random sampling (as done in this video) vs. stratified sampling: www.jstor.org/stable/1268522. A really good read!

@SubZero101010: thanks for the great question. This particular problem was an instance where you could analytically solve for the distribution around the sum of these two random variables (per the discussion with @gokulr8755 in the comments). As mentioned by @andrewevanyshyn1709, seemingly 'simple' problems become quite complex quickly, and so this technique is a nice way to deal with this situation. Furthermore, this approach is a nice way to validate one's intuition and understanding of probability. For example, it is not necessarily obvious that the sum of a series of standard uniform distributions follows the Irwin-Hall distribution. Therefore, I sometimes like to use this technique as a way to validate that my derivations for simpler problems to ensure that they are sound.

My project is to estimate the GPA distribution of my batch through the grading stats of each course that we get. The grade in each course being a random variable and the GPA being a weighted average. Was thinking of convolution approach to do this but now Monte Carlo seems computationally inexpensive compared to it(if it could work for this scenario that is)

Thanks, @gokulr8755, for this really fantastic question. You’ve made a really good point. In this problem, we are summing not just two random variables, but two independent random variables. The perfect case for convolutions to estimate the distribution of A + B. Here is then when I see Monte Carlo as being appropriate: 1. Fairly large, complex problems, where determining the underlying distribution for some application is difficult (or at least fairly involved) to compute. 2. Situations where you are satisfied with an “approximate” answer. From my own personal experience, the latter situation is what we typically find in practice. There are a couple of reasons. One of which is that, in the video problem, the true distributions of A and B are likely unknown. We are likely estimating those distributions from data, but in statistics the true distributions for A and B are not truly known. In addition, I should note that we may use Monte Carlo simulation to learn a “good” set of decision-rules. These days, I mainly work on reinforcement learning problems, where we will (1) simulate different realizations of the future; (2) make decisions in this simulation environment; (3) observe the reward of those decisions; and (4) update our decision-making policies based on the observed rewards. This did not really apply in this problem, as I was only interested in the distribution for a single decision (I.e., do the two reports), but Monte Carlo simulation is quite valuable in this context. I’ll plan on covering this in 3 videos. Lastly, I should note that, depending on the problem, you may be interested in other sampling methods such as Latin Hypercube sampling, which are generally more efficient and require less samples. Really great questions. I sincerely appreciate it.

​@@RiskByNumbers Ah yes I'll look into that professor... It'd be great if you could bring out videos about RL, Markov chains, queuing, simulated annealing etc as these are some really cool topics of applied math in ML. Wanna get some ideas from you for good RL projects too in a video (which involve a bit of coding)...as an undergrad I(and many others) would need to show and test what they learned by doing projects

@aalb1970: on it! I'll invite him for this coming weekend.

Many thanks!

😂 Funny enough, my spouse said roughly the same thing when I showed her the video, @dang4546.

I enjoyed this note, @matejnovosad9152. If you don't mind me asking, for which applications/domain?

​@@RiskByNumbersmany things, mostly verifying my calculations. For example when I was a beginner and was not sure how to calculate the probability of one random variable from Normal dist being greater than another one from a different Normal dist. Or during covid instead of solving the diff equations to find sick people etc. I just simulated it using probabilities and many interations

Hi @alperkaya8919: correct! Perhaps more precisely: there is (approximately) a 12% chance that the 2 reports will take you more than 9 hours to complete. As the party is in 9 hours, there is a 1 - 0.12 = 88% chance that you will finish both reports prior to the party's start time. Thanks for engaging with this content.

@@RiskByNumbers Thank you very much for explaining!!!!!!

Thanks, @kdifiskeicis284kd72j, for engaging with this content! I appreciate it.

Hi Val -- these are all excellent comments and points. With regards to your first comment: good point around the word choice "reasonable". It is making me think that I should put together a short video around risk and probabilities across contexts. As a simple example, engineering structures such as buildings and bridges are designed with a much lower probability of failure than 12-13%. A 12-13% probability of a failure would be considered quite high given the consequences of a failure could be catastrophic. Conversely, a 12-13% chance you are late to a meet-up would likely be viewed as more acceptable, in part because the consequences are much lower. As for your second comment: yes, I can definitely provide a tutorial on other distributions! I similarly received a good question in my video where I covered fitting data to a distribution in Python. I was asked how do we determine which distribution is "appropriate" (e.g., Gaussian, log-normal, etc.). So, I'll provide a tutorial on that topic soon. Feel free to reach out to me via email to continue the conversation!

A video about this would be great too! Im also intrested in more of this kinda of content@@RiskByNumbers

On it@@tito9641 !

@joacimjohnsson6704: there is actually a nice analytical solution to this particular problem. I'd also suggest checking out the Irwin-Hall distribution, which is the distribution for the sum of "n" independent standard uniform random variables: en.wikipedia.org/wiki/Irwin-Hall_distribution. When "n" is large for the Irwin-Hall distribution, indeed it approximately follows a normal distribution. Conversely, if n = 2, we arrive at a probability density that resembles the one in the video. Based on this comment as well as one other in the thread, I am thinking that it may be worthwhile to step through convolutions on this channel. Thanks for the feedback, and definitely reach out to me directly to continue the conversation.

@seyproductions: awesome! As a small comment, you could actually use the random library to model the time to complete each report as both discrete (as you've done above with 'randint') or continuous! To do the latter, you could just change the following 2 lines of code to: first_report = random.random()*(5-1)+1 second_report = random.random()*(6-2)+2 This is inverse transform sampling, which forms the basis of other sampling methods. The idea is that the CDF for our uniform random variables (and any other random variable, for that matter) lies between 0 and 1. The CDF for a uniform distribution for the range between the minimum and maximum is: F(x) = (x-min)/(max-min). We can also rewrite this equation as: x = F(x)*(max-min)+min. Therefore, we use random.random() to generate a random CDF value between 0 and 1 (i.e., F(x)), and map that out to x. It is a pretty neat concept. Interestingly, the probabilities of completing the reports in 9 hours is extremely close for the discrete (88%) vs. continuous (87.5%) approaches (though just slightly different)! Finally, thanks for posting this comment -- really appreciate the effort and thought.

@scalex1882: thanks for the comment. I appreciate that you took the time to (1) view this video; (2) provide feedback; and (3) list clarifications questions that you have after watching it. I'd be happy to expand on this topic with you, perhaps over email. Feel free to reach out at riskbynumbers@gmail.com.

😂😂😂

Yeah, Tod blows

@TheGooner-uh2jx: these are really great points/questions. We actually don't need to define the standard deviation explicitly, as it is implicitly defined already via our distribution. What do I mean by that? The variance for a uniform distribution happens to be 1/12*(max-min)^2. Based on that, the variance for our problem is 16/12 = 4/3 for each report, and the standard deviation is therefore (4/3)^0.5. The above equation can be derived by knowing the following relationship: Var[X] = E[X^2] - E[X]^2. I'm thinking that this is of worthy discussion on the blog that I've just started to put together. Long story short, when you define any parametric distribution, the parameters selected define the variance implicitly. For certain distributions, such as the normal distribution, you need to define the standard deviation/variance given that it defines its probability density. Let me know if that helps, and feel free to reply with further questions!

@@RiskByNumbers Thanks for the reply. I didn't realize your example was Uniform Distribution (didn't even know what that was until now). In my past experiences we mostly assumed Normal Distribution in Minitab. Excited to learn from your blog! I'm a mechanical engineer with very basic statistics background. I have a lot to learn!