More on expected value
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Law of large numbers
Let's learn a little bit about the law of large numbers, which is on many levels, one of the most intuitive laws in mathematics and in probability theory. But because it's so applicable to so many things, it's often a misused law or sometimes, slightly misunderstood. So just to be a little bit formal in our mathematics, let me just define it for you first and then we'll talk a little bit about the intuition. So let's say I have a random variable, X. And we know its expected value or its population mean. The law of large numbers just says that if we take a sample of n observations of our random variable, and if we were to average all of those observations-- and let me define another variable. Let's call that x sub n with a line on top of it. This is the mean of n observations of our random variable. So it's literally this is my first observation. So you can kind of say I run the experiment once and I get this observation and I run it again, I get that observation. And I keep running it n times and then I divide by my number of observations. So this is my sample mean. This is the mean of all the observations I've made. The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. Or I could also write it as my sample mean will approach my population mean for n approaching infinity. And I'll be a little informal with what does approach or what does convergence mean? But I think you have the general intuitive sense that if I take a large enough sample here that I'm going to end up getting the expected value of the population as a whole. And I think to a lot of us that's kind of intuitive. That if I do enough trials that over large samples, the trials would kind of give me the numbers that I would expect given the expected value and the probability and all that. But I think it's often a little bit misunderstood in terms of why that happens. And before I go into that let me give you a particular example. The law of large numbers will just tell us that-- let's say I have a random variable-- X is equal to the number of heads after 100 tosses of a fair coin-- tosses or flips of a fair coin. First of all, we know what the expected value of this random variable is. It's the number of tosses, the number of trials times the probabilities of success of any trial. So that's equal to 50. So the law of large numbers just says if I were to take a sample or if I were to average the sample of a bunch of these trials, so you know, I get-- my first time I run this trial I flip 100 coins or have 100 coins in a shoe box and I shake the shoe box and I count the number of heads, and I get 55. So that Would be X1. Then I shake the box again and I get 65. Then I shake the box again and I get 45. And I do this n times and then I divide it by the number of times I did it. The law of large numbers just tells us that this the average-- the average of all of my observations, is going to converge to 50 as n approaches infinity. Or for n approaching 50. I'm sorry, n approaching infinity. And I want to talk a little bit about why this happens or intuitively why this is. A lot of people kind of feel that oh, this means that if after 100 trials that if I'm above the average that somehow the laws of probability are going to give me more heads or fewer heads to kind of make up the difference. That's not quite what's going to happen. That's often called the gambler's fallacy. Let me differentiate. And I'll use this example. So let's say-- let me make a graph. And I'll switch colors. This is n, my x-axis is n. This is the number of trials I take. And my y-axis, let me make that the sample mean. And we know what the expected value is, we know the expected value of this random variable is 50. Let me draw that here. This is 50. So just going to the example I did. So when n is equal to-- let me just [INAUDIBLE] here. So my first trial I got 55 and so that was my average. I only had one data point. Then after two trials, let's see, then I have 65. And so my average is going to be 65 plus 55 divided by 2. which is 60. So then my average went up a little bit. Then I had a 45, which will bring my average down a little bit. I won't plot a 45 here. Now I have to average all of these out. What's 45 plus 65? Let me actually just get the number just so you get the point. So it's 55 plus 65. It's 120 plus 45 is 165. Divided by 3. 3 goes into 165 5-- 5 times 3 is 15. It's 53. No, no, no. 55. So the average goes down back down to 55. And we could keep doing these trials. So you might say that the law of large numbers tell this, OK, after we've done 3 trials and our average is there. So a lot of people think that somehow the gods of probability are going to make it more likely that we get fewer heads in the future. That somehow the next couple of trials are going to have to be down here in order to bring our average down. And that's not necessarily the case. Going forward the probabilities are always the same. The probabilities are always 50% that I'm going to get heads. It's not like if I had a bunch of heads to start off with or more than I would have expected to start off with, that all of a sudden things would be made up and I would get more tails. That would the gambler's fallacy. That if you have a long streak of heads or you have a disproportionate number of heads, that at some point you're going to have-- you have a higher likelihood of having a disproportionate number of tails. And that's not quite true. What the law of large numbers tells us is that it doesn't care-- let's say after some finite number of trials your average actually-- it's a low probability of this happening, but let's say your average is actually up here. Is actually at 70. You're like, wow, we really diverged a good bit from the expected value. But what the law of large numbers says, well, I don't care how many trials this is. We have an infinite number of trials left. And the expected value for that infinite number of trials, especially in this type of situation is going to be this. So when you average a finite number that averages out to some high number, and then an infinite number that's going to converge to this, you're going to over time, converge back to the expected value. And that was a very informal way of describing it, but that's what the law or large numbers tells you. And it's an important thing. It's not telling you that if you get a bunch of heads that somehow the probability of getting tails is going to increase to kind of make up for the heads. What it's telling you is, is that no matter what happened over a finite number of trials, no matter what the average is over a finite number of trials, you have an infinite number of trials left. And if you do enough of them it's going to converge back to your expected value. And this is an important thing to think about. But this isn't used in practice every day with the lottery and with casinos because they know that if you do large enough samples-- and we could even calculate-- if you do large enough samples, what's the probability that things deviate significantly? But casinos and the lottery every day operate on this principle that if you take enough people-- sure, in the short-term or with a few samples, a couple people might beat the house. But over the long-term the house is always going to win because of the parameters of the games that they're making you play. Anyway, this is an important thing in probability and I think it's fairly intuitive. Although, sometimes when you see it formally explained like this with the random variables and that it's a little bit confusing. All it's saying is that as you take more and more samples, the average of that sample is going to approximate the true average. Or I should be a little bit more particular. The mean of your sample is going to converge to the true mean of the population or to the expected value of the random variable. Anyway, see you in the next video.