[MUSIC] So today we'll be talking about the expected value. What we mean by that is, we will be looking at a trial, when we conduct a certain trial. And we will be looking at the outcomes that we see from this trial. And the question that we are interested in is, what is the expected value that we expect to see? So, for example, if I am rolling a dice just once and it is a fair dice with 6 sides, what is the value that do I expect once I roll that dice? Okay, so, that question becomes actually interesting, because what is the meaning of that value? So, I have one dice and I roll it, the possible outcome that I can see are one, two, three, four, five, six, these are the possible values. So now I'm saying, if I roll a dice once, what is the expected value? So each one of these since the dice is, the dice is fair, each one of these is going to have a probability of one over six. All right, it's one over six. And the question is, what does it mean to say that I expect to see a certain value when I roll that dice? One interpretation of that is that, is, is to think about the following trial or experiment. If I take that same dice and I roll it a very large number of times. Think about it, roll it for a million times, a billion times, and so on. And then I look at the values that I've seen every time. And then I look at the average value from these values. And look at the, what do I see there? So for example, I take this, I take this dice, I roll it. Let's make our, our life easier and let's say, we roll it 600 times. So I wrote 600 times, so and every time I roll it, I write down the, the side I see, the number I see. So this is the first time. This is the second time, the third time, all the way to 600. So if I ask another question, is that for this dice since it is fair. It has six sides, each one with the, appearing with the probability one over six. And I ask you, what do you expect to see here as number in these boxes when I'm writing down these values 600 times. I think it makes sense to say that we are going to see 1 on the order of about 100 times. We are going to see 2 on the order of 100 times, 3 on the order of 100 times all the way to 6 of the order 100 times, right? If the, if the probability of seeing 1 is 1 over 6. So, but what this mean is that we expect to see 1 every 6 times we roll that dice. So here, now if I am saying what is the average value that I expect to see when I roll this dice 600 times? I would basically say, okay I have, I've seen the value of 1 100 times. I've seen the value 2, 100 times. I have seen the value 3 100 times. I have seen the value 4, 100 times. I have seen the value 5, 100 times and the value 6,100 times. I look at this and all this divided by 600, and this is going to give me the expected value or that I would see from rolling this dice 600 times, or similarly this would be the expected value of seeing each one, the, the expected value of that I would see when I roll that dice once. Okay? So in this case, what we are saying that we basically take the value that we see times the number of times we have observed. We take the sum over all these possibilities, divide by the number of time we repeated that same experiment. Now going back to the history of expected value and looking at this dice. What is the expected value that I would observe if I roll the dice only once? Okay. So the way would do that again in this case is 1, the value 1 will appear with probability of 1 over 6. 2 with the probability 1 over 6 or 3 with the probability 1 over 6 plus 4 with the probability 1 over 6. Okay. So, this would be the expected value. When, if I roll this dice only once. And if we compute this value here, so we have 1 6th plus 2 6th plus 3 6th plus 4 6th plus 5 6th plus. 1 plus 2, 3, 6, 10, 15, 21 over 6, and this is 3.5. So this is the expected value for this trial where I roll this six sided fair dice once. So I expect to see a value of 3.5. So this immediately should raise a flag. What does it mean to say that the expected value is 3.5? This dice actually does not have a side that has a value of 3.5, okay? So this is why it makes sense to think about the meaning of this notion of expected value in terms of the mean or the average value that we would see if we had roll this dice a very large number of times. Mathematically in, in fact is that, if we roll it an infinite number of times, okay? So this will be the expected value of this dice. Now if we make this a bit more interesting and I say, I take this dice and I roll it twice. Okay. So I take the dice, roll it once. I roll it twice, a second time. And then I record the, the two values that I have seen. And I look at the sum of these two values. And then I say, what is the expected value I would see. The expected sum, of the two numbers I get from rolling that dice twice. Okay. So now, the first question we have to ask, is what are the possible sums that we can see? Since we have the dice we are rolling it twice, and we are recording the two numbers. So we have two numbers, x, y, right? So we are going to roll the dice first and we are going to see some value x. We roll it again, we see some value y. Of course, x can be either 1 or 2 or 3 or 4 or 5 or 6. The same thing with y. So if we look at the sum of these two, we ask what is the possible sum we can observe? It going to be, the smallest possible is going to be to where both rows basically showed 1, or it can be 3, can be 4, all the way to the largest possible sum which is 12. Now, when we compute the, the expected sum from this experiment or this trial, we basically have to see what are possible ways of generating each one of these sums. So for example x plus y equal 2, to, to get the sum 2, x has to be 1 and y has to be 1. So the first time I rolled the dice, I've seen 1, second time I have seen 1. There is no other option for getting a sum of 2 given this dice that I'm rolling it twice. Now what is the probability of seeing 1 the first time? The answer is 1 over 6. What is the probability that I see 1 the second time? It is 1 over 6. Now since these two, two, two events are independent, the probability of seeing 1 and 1 is 1 over 6 times 1 over 6 and the probability is 1 over 36. This is for the sum of 2. What happens if the sum is 3? The, the, the way I could have gotten a sum of three, is either x is 1 the first time I rolled the dice I saw 1, the second time it's 2, or the first time I saw a 2 and the second time I saw a 1. Again the probability of seeing 1, and then seeing 2 is 1 over 6 time 1 over 6, 1 over 36. And this is 1 over 6, 1 over 6 equal 1, 36. So the probability of seeing this is this plus this area, right? So it is. Okay. And then again we can get what is the probability, for example, that x plus y is 4. And we just have to list all the possibilities, which are either x is, the first time I rolled the dice it was, I saw 1. Then I saw 3, or the first time it's 2 and the second time is 2 or the, sorry this is x. Or the first time is 3 and the second time is 1. And using similar, computation like this, the probability of this happening, this event, 1 and 3 is 1 over 36, 1 over 36, 1 over 36. So the probability that the sum is 4 is 3 over 36. We can re-repeat this all the toward 12, what is the probability of seeing a sum of 12? Again, in this case, the only possibility is seeing 6 first time and seeing 6 the second time. We end up with a probability of 1 over 36. So now we can say what is the expected value of the sum? What is the expected sum that we, okay, we would see from this trial where we roll the dice, the dice twice? It's basically the probability of observing any of the sums. I or we can say probability of whatever variable of x plus y times x plus y. And we sum over all possible values of xy, x plus y equal 2, 2 16th. So this notation here is saying that the sum x plus y can be 2 or 3 or 4 all the way to, to 12, sorry, not 16. And what is the probability of observing each sum times the sum itself. Which is in our case, is the probability of 2 times 2, plus the probability of the sum being 3 times 3, the probability of 4 times 4, the probability observing 12 times 12. We have seen that the probability that the sum is 2 is 1 over 36 times 2 plus the probability of, that see we, we have sum of 3 is 2 over 36 times 3. The probability of 12 again is 1 over 36 times 12, and this will give us the expected value of that sum. So, now this is again the expected value for rolling a dice twice and looking at the sum and so on. But, there are even more, much more interesting applications of this expected value and what, what kind of value we expect to see when we have a [UNKNOWN] process or a random process. For example, in the case of DNA sequences, if I am having 2 sequences, A-C-C-T and A-C-G-T, and I am saying that this sequence evolved from this sequence. For after a given amount of time, the question is some observe these two sequences and they ask are these two sequences the same? Or do they correspond? Of course if I compare these two sequences, I see they differ at the third position, right? But the question is, given that we know that mutations happen, that they change the letters, these DNA nucleotides, is this something that I would expect to see? So for example, if I know that there's a probability of .25, a probability of one quarter, that the position would change in the sequence, okay? And every position is independent, then this is expected after a certain amount of time, because if I'm saying that every letter here has a probability of 1 over 4 of changing to a different letter, then I would expect this after certain amount of time, right? Because mutations will happen. I wouldn't, I wouldn't expect this if not time has past, the sequence stays the same, but if time has passed and mutation can happen, then we have this notion of expectation or what is the expected sequence after certain amount of time and knowing something about mutation. So, in this case the notion of expectation, and the expected value is central to us being able to say whether this sequence and this sequence are related, versus are they different because they have nothing to do with each other.
