Learn about Expected Value: E- X Video

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Very nice and useful explanation.

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Learn about Expected Value: E- X

When we first started talking about central tendencies and how we measure average, we talked about the arithmetic mean and there you just add it up the numbers and you divided by the number of numbers there were. So let's say our population of number is—say we have a 3, 3, 3, 4, and 5 that’s our population and if we wanted the population mean here, we would just add all the numbers up, we would say 3+3+3+4+5 and then we would divided that sum by the number of numbers we have. We would divide that by 5 and we would come out—0:42 this would 9+9 would be 18 over 5. That would 18/5 which would be what, 3 and 3/5 which is 3.6. That’s the population mean for this population of numbers. If we just rearrange the map here a little bit, we could view it slightly different way. How many 3’s do we have? We have three 3’s so we could view this as 3(3). How many 4’s do we have? We have one 4 so it's plus, 1(4) + 1(5), all of that divided by 5. And then what we could do is, this is the same thing and I’m just doing a little bit of basic really number manipulation here. This is a same thing as 1/5 (3.3 + 1.4 + 1.5) and then if we distribute this 1/5 this is equal to what? 3/5.3 + 1/5.1 + 1/5.4 + 1/5.5 and this were we can see a bunch of different ways. Let me express this as decimals. So 3/5 that’s—so it's .6 x 3 + .2 x 4 + .2 x 5 or we can then express this decimal as percentage. We could say 60% x 3 + 20% x 4 + 20% x 5. This is identical to adding up the numbers and then dividing by the total number of numbers there are, right? But this is interesting. Because here, we had to know how many total numbers there are. We had to—we added up 5 numbers, we divided by 5. All I did is I change the—I played around with the arithmetic a little bit and I got to this expression. But these expressions more interesting or at least it's different. It's not necessarily more interesting. I don’t want to make any value judgment about here. But here, I don’t know how many numbers there are. I’m just telling you about the frequency of the numbers. I’m telling you that 60% of the numbers are 3, 20% numbers are 4 and then 20% of the numbers are 5. And then if I were to calculate this out I would get 60% x 3 is 1.8 + 20% x 4 is 0.8 + 20% x 5 is—that’s the 20% +1 would should be equal to 2.6 + 1 is equal to 3.6. So we would get the exact same number. But what’s interesting here is that this tells you just a frequencies really the relative to the frequencies of the 3’s, the 4’s and 5. What percentage of this population is 3’s 60%? What percentages 4’s 20% and what population is 5’s? And I’m doing that because we’ve just talked about random variables and all of that and in the beginning we start our statistic discussion about populations and samples but if you think about it, every time you get kind of—you do one of your experiments and you get new value for random variable. Alright, let say our—let's go to our classroom example. We have our random variable x equal to—I don’t know. It's equal to the number of heads after six tosses of fare coin so that’s our random variable. So hopefully now we could kind of connect what we’ve thought about in terms of just arithmetic mean and central tendency and population versus sample and then connect that to the notion of our random variables. So when we first start talking about statistics, we said, okay you have this notion of a population and then you would sample the population and we gave them a couple of examples. The most common one is you wanted to predict the out come of our presidential election, the population is everyone who’s going to vote on the election. You can't survey all 15 million people or whatever who’s going to vote for the election. So what you do is you survey a random sample of that population and then you can calculate statistics on that sample then hopefully can estimate the population as a whole. But what happens if the population is not finite? And just to go back, if you the population is finite, you can calculate things like the population mean, right. We learn the population mean was that µletter and that was just you really take up all of the items and the population, add them together and you divide whether the number of items there are. That’s what we did up here. If this was a whole population of numbers, we’ve figure it out that this was µ. If this was a sample from a population then this would be the sample mean but we learned all about that but that’s not what I want to get it now. But what happens if this population is infinite? If it's infinite and you’re like, oh sounds that it doesn’t make any sense. But if you think about it, well a random variable really is the population it's—you can kind of notice each instance of a random variable or every time you performed the experiment, you’re taking out an instance of an infinite population, right? You can perform this experiment in infinite number of times. You can just keep doing it. It's not like after doing the thousand times, you’re like, oh you can toss six coins six times anymore and count the number of heads. You can then perform this indefinitely, right? So every specific result from a random variable and those are usually lower case results. Lower case x1 or x2 or x3, these are just specific instances of a random variable. You view this as samples from an infinite population. So I’ll try to draw an infinite population, it's kind of harder. Maybe I’ll draw arrows that go often every direction. This population never ends. You can keep performing the experiment and keep getting samples. But your sample is usually finite and this would be our—let's say we perform this experiment. We’ve toss a fare coin six times and we do that experiment—I don’t know. We do it a hundred times, so then we would have a hundred samples, x2, I don’t go all the way x 100. And the reason why I’m doing this connection is one, to make you see the connection between the random variable and the probability and the statistics that we’ve talked about earlier and in this video, I want to introduce you to the concept of the expected value of a random variable and it's nothing else. So the expected value of a random variable is the exact same thing as the population mean effects. Sometimes, it's called the population mean but what it makes interesting is, in this situation, you have an infinite population, right. So you can't just add up all the numbers and divide it by the numbers you have because you have an infinite number of numbers. But what you can do is if you say, wow, do I know the frequency of the numbers, right? I know that 3 shows up 60% of the time, 4 shows up 20% of the time, 5 shows up 20% of the time then even if you have an infinite number of numbers, you can actually still calculate a mean and that’s how you do it for an expected value for of a random variable. So how do you figure out the frequencies that number show up? Well you can look at the probably of distribution, the discreet probability of distribution. So let's say on that example that we did the last time, I forgot the exact numbers but actually let me just take what we did, our Excel out—I know. Let me just quickly, I want to make N 6 trials proper probability of heads, tails, okay .5 and then I need to change what this chart. I just give me one second. Change with the inputs of these chart are—let me just off the screen right now. Okay, there you go. So this is a probability of distribution for what I just describe. I have a fare toss of a coin and I want to know how many heads I have after six tosses, right. So you can perform this experiment a bunch of times but this tells you the frequency, the frequency of how of that random variable. So when you perform that experiment, let see, whatever does it, 0.09% of the time, or not actually—9% of the time, you’re going to get exactly one head. If you look 23% of the time, you’re going to get 2 heads. 31% of the time, you’re going to get 3 heads. 23% of time, you’re going to get 4 heads and then 9% of the time, you get a 5 heads. And then 2% of time, you’re going to get 6 heads. So if you have that information, you can then actually figure out the population mean for this population as describe by this probability of distribution or the expected value and let's do that right here. I’ll put this over to the side. So I’m looking at that chart I just did while I do this. So if this—we just look at the probability distribution for this random variable, the number of heads after 6 tosses of fare coin. So the expected value of our random variable is going to be each outcome—so the first outcome is x that we had zero heads times the frequency that zero shows up. So we figure out before that the frequency—it's a little inexact so I don’t have—actually I have the exact numbers. Let me— The frequency—so we get zero will show up in our random variable, .01563% of the time, so let me write that. So we’re going to say, will this happens and I can write as a percentage, 1.563 percentages of the time plus one happens 9.375% of the time and then plus 2 happens 23.438 percentage of the time plus 3 happens, let's see. It says 31.25% of the time, almost there. I get 4 heads out of six tosses, 23% of the time times 23.438%, I get 5 heads after six tosses, 9.375% of the time and finally I get all heads 1.563% of the time. 1.563 and that makes sense again because all heads should be just as likely as all tails right? All tails is the same things as no heads. So what we did here is exactly what we did up here. We took the relative frequency of each of the numbers in the population and we multiply that outcome times it's relative frequency or adding it up. But this is the exact same thing mathematically as we did up here. But what's useful now is we can apply the same principles but we’re finding the arithmetic mean of an infinite population or the expected value of a random variable which is the same thing as the arithmetic mean of the population of this random variable. So this value would be equal to—actually let me just use Excel to calculate it. So the expected value the number of heads you get after six tosses, so this will be—see I want to do, so this is zero times it's frequency and then I’m going to add them all up and then this will just do that same thing all of it. So this says this will be one time. It's frequency 2 times it's frequency and then if I were going to take the sum of all of them equal sum of all of this, I get—let me add some. I get exactly 3. So that’s it, and that’s actually kind of an expected outcome right? Well I shouldn’t use the word expected too much. That the central tendency or you could say the population mean of this random variable, you could say the expected value of this random variable is exactly 3 and that in this example, it turned out that 3 is also in kind of the colloquial sense, it's the most expected value right. It's the most probable value but we’ll see in the future that the expected value does it have to be the most probable value. You could have a very probability of having no heads and a very high probability of having six heads and then you’d still have an expected value of 3 even if six or zero are more problem and I show you more examples of that. But the purpose of this video is to really show you that the expected value calculation is the same thing as the population mean calculation but we do is this way because you can't add up an infinite number of data points and divide it by an infinite number. Instead, you want to know the frequencies of each of the outcomes and then you just add up all of the outcomes awaited by their frequencies but that’s no different than what you did up there. And I really want to hit that point home because sometimes in probability books, they’ll just give you a formula of the expected variable distribution as each of the outcomes times their frequency. But I want to show you that that is the same thing as the population mean. Anyway, see you in the next video.

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When we first started talking about central tendencies and how we measure average, we talked about the arithmetic mean and there you just add it up the numbers and you divided by the number of numbers there were. So let's say our population of number is—say we have a 3, 3, 3, 4, and 5 that’s our population and if we wanted the population mean here, we would just add all the numbers up, we would say 3+3+3+4+5 and then we would divided that sum by the number of numbers we have.

We would divide that by 5 and we would come out—0:42 this would 9+9 would be 18 over 5. That would 18/5 which would be what, 3 and 3/5 which is 3.6. That’s the population mean for this population of numbers. If we just rearrange the map here a little bit, we could view it slightly different way. How many 3’s do we have? We have three 3’s so we could view this as 3(3). How many 4’s do we have? We have one 4 so it's plus, 1(4) + 1(5), all of that divided by 5.

And then what we could do is, this is the same thing and I’m just doing a little bit of basic really number manipulation here. This is a same thing as 1/5 (3.3 + 1.4 + 1.5) and then if we distribute this 1/5 this is equal to what? 3/5.3 + 1/5.1 + 1/5.4 + 1/5.5 and this were we can see a bunch of different ways. Let me express this as decimals. So 3/5 that’s—so it's .6 x 3 + .2 x 4 + .2 x 5 or we can then express this decimal as percentage. We could say 60% x 3 + 20% x 4 + 20% x 5. This is identical to adding up the numbers and then dividing by the total number of numbers there are, right?

But this is interesting. Because here, we had to know how many total numbers there are. We had to—we added up 5 numbers, we divided by 5. All I did is I change the—I played around with the arithmetic a little bit and I got to this expression. But these expressions more interesting or at least it's different. It's not necessarily more interesting. I don’t want to make any value judgment about here.

But here, I don’t know how many numbers there are. I’m just telling you about the frequency of the numbers. I’m telling you that 60% of the numbers are 3, 20% numbers are 4 and then 20% of the numbers are 5. And then if I were to calculate this out I would get 60% x 3 is 1.8 + 20% x 4 is 0.8 + 20% x 5 is—that’s the 20% +1 would should be equal to 2.6 + 1 is equal to 3.6. So we would get the exact same number.

But what’s interesting here is that this tells you just a frequencies really the relative to the frequencies of the 3’s, the 4’s and 5. What percentage of this population is 3’s 60%? What percentages 4’s 20% and what population is 5’s? And I’m doing that because we’ve just talked about random variables and all of that and in the beginning we start our statistic discussion about populations and samples but if you think about it, every time you get kind of—you do one of your experiments and you get new value for random variable.

Alright, let say our—let's go to our classroom example. We have our random variable x equal to—I don’t know. It's equal to the number of heads after six tosses of fare coin so that’s our random variable. So hopefully now we could kind of connect what we’ve thought about in terms of just arithmetic mean and central tendency and population versus sample and then connect that to the notion of our random variables.

So when we first start talking about statistics, we said, okay you have this notion of a population and then you would sample the population and we gave them a couple of examples. The most common one is you wanted to predict the out come of our presidential election, the population is everyone who’s going to vote on the election. You can't survey all 15 million people or whatever who’s going to vote for the election. So what you do is you survey a random sample of that population and then you can calculate statistics on that sample then hopefully can estimate the population as a whole.

But what happens if the population is not finite? And just to go back, if you the population is finite, you can calculate things like the population mean, right. We learn the population mean was that µletter and that was just you really take up all of the items and the population, add them together and you divide whether the number of items there are. That’s what we did up here. If this was a whole population of numbers, we’ve figure it out that this was µ.

If this was a sample from a population then this would be the sample mean but we learned all about that but that’s not what I want to get it now. But what happens if this population is infinite? If it's infinite and you’re like, oh sounds that it doesn’t make any sense. But if you think about it, well a random variable really is the population it's—you can kind of notice each instance of a random variable or every time you performed the experiment, you’re taking out an instance of an infinite population, right?

You can perform this experiment in infinite number of times. You can just keep doing it. It's not like after doing the thousand times, you’re like, oh you can toss six coins six times anymore and count the number of heads. You can then perform this indefinitely, right? So every specific result from a random variable and those are usually lower case results. Lower case x1 or x2 or x3, these are just specific instances of a random variable. You view this as samples from an infinite population. So I’ll try to draw an infinite population, it's kind of harder. Maybe I’ll draw arrows that go often every direction.

This population never ends. You can keep performing the experiment and keep getting samples. But your sample is usually finite and this would be our—let's say we perform this experiment. We’ve toss a fare coin six times and we do that experiment—I don’t know. We do it a hundred times, so then we would have a hundred samples, x2, I don’t go all the way x 100.

And the reason why I’m doing this connection is one, to make you see the connection between the random variable and the probability and the statistics that we’ve talked about earlier and in this video, I want to introduce you to the concept of the expected value of a random variable and it's nothing else. So the expected value of a random variable is the exact same thing as the population mean effects. Sometimes, it's called the population mean but what it makes interesting is, in this situation, you have an infinite population, right. So you can't just add up all the numbers and divide it by the numbers you have because you have an infinite number of numbers. But what you can do is if you say, wow, do I know the frequency of the numbers, right?

I know that 3 shows up 60% of the time, 4 shows up 20% of the time, 5 shows up 20% of the time then even if you have an infinite number of numbers, you can actually still calculate a mean and that’s how you do it for an expected value for of a random variable.

So how do you figure out the frequencies that number show up? Well you can look at the probably of distribution, the discreet probability of distribution. So let's say on that example that we did the last time, I forgot the exact numbers but actually let me just take what we did, our Excel out—I know. Let me just quickly, I want to make N 6 trials proper probability of heads, tails, okay .5 and then I need to change what this chart. I just give me one second. Change with the inputs of these chart are—let me just off the screen right now.

Okay, there you go. So this is a probability of distribution for what I just describe. I have a fare toss of a coin and I want to know how many heads I have after six tosses, right. So you can perform this experiment a bunch of times but this tells you the frequency, the frequency of how of that random variable. So when you perform that experiment, let see, whatever does it, 0.09% of the time, or not actually—9% of the time, you’re going to get exactly one head. If you look 23% of the time, you’re going to get 2 heads.

31% of the time, you’re going to get 3 heads. 23% of time, you’re going to get 4 heads and then 9% of the time, you get a 5 heads. And then 2% of time, you’re going to get 6 heads. So if you have that information, you can then actually figure out the population mean for this population as describe by this probability of distribution or the expected value and let's do that right here. I’ll put this over to the side. So I’m looking at that chart I just did while I do this.

So if this—we just look at the probability distribution for this random variable, the number of heads after 6 tosses of fare coin. So the expected value of our random variable is going to be each outcome—so the first outcome is x that we had zero heads times the frequency that zero shows up. So we figure out before that the frequency—it's a little inexact so I don’t have—actually I have the exact numbers. Let me—

The frequency—so we get zero will show up in our random variable, .01563% of the time, so let me write that. So we’re going to say, will this happens and I can write as a percentage, 1.563 percentages of the time plus one happens 9.375% of the time and then plus 2 happens 23.438 percentage of the time plus 3 happens, let's see. It says 31.25% of the time, almost there. I get 4 heads out of six tosses, 23% of the time times 23.438%, I get 5 heads after six tosses, 9.375% of the time and finally I get all heads 1.563% of the time. 1.563 and that makes sense again because all heads should be just as likely as all tails right? All tails is the same things as no heads.

So what we did here is exactly what we did up here. We took the relative frequency of each of the numbers in the population and we multiply that outcome times it's relative frequency or adding it up. But this is the exact same thing mathematically as we did up here. But what's useful now is we can apply the same principles but we’re finding the arithmetic mean of an infinite population or the expected value of a random variable which is the same thing as the arithmetic mean of the population of this random variable.

So this value would be equal to—actually let me just use Excel to calculate it. So the expected value the number of heads you get after six tosses, so this will be—see I want to do, so this is zero times it's frequency and then I’m going to add them all up and then this will just do that same thing all of it. So this says this will be one time. It's frequency 2 times it's frequency and then if I were going to take the sum of all of them equal sum of all of this, I get—let me add some. I get exactly 3. So that’s it, and that’s actually kind of an expected outcome right? Well I shouldn’t use the word expected too much.

That the central tendency or you could say the population mean of this random variable, you could say the expected value of this random variable is exactly 3 and that in this example, it turned out that 3 is also in kind of the colloquial sense, it's the most expected value right. It's the most probable value but we’ll see in the future that the expected value does it have to be the most probable value.

You could have a very probability of having no heads and a very high probability of having six heads and then you’d still have an expected value of 3 even if six or zero are more problem and I show you more examples of that. But the purpose of this video is to really show you that the expected value calculation is the same thing as the population mean calculation but we do is this way because you can't add up an infinite number of data points and divide it by an infinite number.

Instead, you want to know the frequencies of each of the outcomes and then you just add up all of the outcomes awaited by their frequencies but that’s no different than what you did up there. And I really want to hit that point home because sometimes in probability books, they’ll just give you a formula of the expected variable distribution as each of the outcomes times their frequency. But I want to show you that that is the same thing as the population mean.

Anyway, see you in the next video.

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