[SOUND] Before we discuss expectations of random variables, we will discuss the notion of average. So what is an average salary in some countries? Okay, this is the total salary of all population divided by the number of employees in this country. So this is the standard notion of average, which is used everywhere. In mathematics it is called arithmetic mean. Okay, let's consider a full in example. We have a student and he had three test, and he got scores 78, 72, and 87. What is his average score on these tests? Okay, to compute average we need to do the following. We need to add all these scores and divide by their number. So we have to add 78, 72 and 87, divide by 3, there are 3 tests. So we have 79. Note that in this case we got lucky and the answer is integer, in general, this is not guaranteed. For example, if you use instead of 87 you consider 86 there, you will not obtain an integer number. We will have 78 and two-thirds. So this is not always the case but in this case we have an integer number. Let's consider one more example related to the notion of average. Suppose we have a company and there is HR management in this company, and they use the following strategy. They look at everyone's performance and they fire everyone who performs below average. So what will be a result of such a strategy? Okay, this might sound reasonable if someone who works below average probably the company should fire him. Okay, but note that unless everyone works equally, which is an extremely rare case, it's very rare that two persons perform equally, if you measure their work by some number. If it doesn't happen, if it's not the case that everyone works equally, then there is always someone who works below average. Average is something in the middle, so someone works better, someone works not that good, and so there is someone below average because average is somewhere in the middle. So, if we fire them, then the average performance will grow. We fired people who work below average, so the average will grow. And then we will have new people below average because if, everyone doesn't work equally then there is someone below average. If we repeat it several times we will just be left with just one person who was the best employee. But is it good that we only have one person left? Okay, let's move on to problems that are more related to random variables. Suppose we have a dice and they throw this dice many times. And each time we have some outcome, what is the average outcome of all of these throws? Okay, can we give a precise answer? No, we cannot because this is a random variable. If each time, you have as a result 6, then the average outcome will be 6. This is quite improbable, but this is possible. If on the other hand, each time you have 1, then the average will be also 1. This is again improbable but this is also possible. So there is no possibility to give some precise answer which will be always the case. But what we can do is we can give an approximation to an average outcome that will be a good approximation with high probability. Okay, let's see what we can do. Suppose we through a dice n times and n is very large. Then among outcomes there are approximately n/6 ones, n/6 twos and so on. So each outcome has probability 1/6, so in approximately 1/6 fraction of all cases we will have one, in approximately 1/6 fraction of all cases we will have two, and so on. Then now we are ready to compute the average. We have to sum up all of the results. So, we have 1/6 x 1. So, we have one n/6 times, we have n/6 x 2 and so on. So we sum all of this up, we can move n outside the brackets and then we have 3.5n. This is the sum of all results, approximately again. Okay, and now to obtain the average we have to divide by the number of experiments. We have to divide by n. We threw the dice n times, so we have to divide by n. So the average is approximately 3.5. Okay, and this is called an expected value or an expectation of a dice throw. [SOUND]