 In this lecture, we review some basic high school math concepts that you'll find that you'll need in our statistics course. Okay, as you can see that symbol there, sigma, that's capital sigma, and we use that to denote some summations. You'll see an Excel too, by the way. So in this example, we summed x1 plus x2 all the way through x7. And notice how simple it is when you just put a summation in front of the xi. Here's how we use the summation symbol in statistics. Note that the sample mean is the sum of all the values, sum of xi, divided by n. Also, we're going to talk about summing squared observations in the course. In the course, we're going to learn about ways to measure variation. One of them is the sample variance, where we're basically getting the deviations around the mean, squaring them, adding them up, and dividing by n minus one. We'll worry about why we're dividing by n minus one and not n once we actually get to the course. So here's just the way we worked out the problem. Remember, the numerator is the deviation from the mean, and square it. So that's one minus three squared, two minus three squared, and so on. When you add all of the squared deviations up, you get 10 for the numerator, and then you divide by n minus one, remember, not by n. We're dividing by four. And so we get a variance of 2.5. When you get to the point where you want to compute the standard deviation, another measure of deviation, you just take the variance and take the square root of that, and that's the standard deviation. Again, more later in the course. Here we'll learn about a weighted average. Suppose you take three exams, but they're not equally weighted. The first exam is worth 20% of the grade. Exam two is worth 30% of the course grade, and the final is worth 50% of the grade. Essentially, your weights are 20, 0.20, 0.30, and 0.50. And now we have one student whose grades were 100, 80, and 90. So how do we compute the average? Okay, we set it up in a table so you can see the grade, the weights, and you can see the weighted average is now an 89. Notice if you use the symbols, it's the sum of the WI, XI. WI represents the weights. And notice the average again is 89. You'll see it's not the same as a simple average. The table was nice and made things kind of simple, but it's not very compact. Using summation notation for the weighted average is actually easier to do. Take a look, you have the weights, you have the grades. Turns out to be 89, exactly the same as what we had before. Here's a question. What formula would we use if we had all the same weights for the grades, if we were weighing them all the same in order to come up with a course average? Next slide, we'll see that. Here we showed that we could use the same formula for a weighted average for the straight average, because if we have three exam grades and they all have the same weight, we're talking about one third, one third, one third. It turns out that that average would be 90, not 89 as we had before when the final exam was worth more. Using one third, one third, one third as the weights is exactly the same as adding up all the grades and dividing by three. And if you can't yet see that it's exactly the same, you should really go back before the course starts and review some of the basic math that you see here. You may have already learned the expected value in another course, and now that you see weighted average using summation notation, I think you can pretty clearly see that expected average is nothing more than a weighted average where the weights are the probabilities. And the expected value is the long run average value of a random variable. In other words, it's the mean of the random variable. It's a weighted average of all the possible values that this random variable can take on, but it's a weighted average. Now, in the next slide, we're going to be looking at an example of a business deal. Let's look at these two business deals. Business opportunity one, you have a 20% chance of making a million, a 30% chance of making 500,000, and a 50% chance of making zero. Well, you might analyze it through expected values. See the way we symbolize expected value, e, the x in parentheses, that's called the expected value of x. And it turns out for this problem, the expected value of x is $350,000. If you were able to do this business deal over and over again, you average out making $350,000. Of course, you do it once, you're not making $350,000, either making a million, $500,000 or zero. But the expected value, which is a long run kind of probability, this is what you'd expect to make. You'd be keeping doing this over and over again, $350,000. Business opportunity two, where there's a 15% chance of making 10 million, a 10% chance of making 5 million, and a 75% chance of making zero, the expected value turns out to be $2 million. When you toss a die, this is going to shock you. What is the expected value? Now, you can get either a one, a two, a three, a four, a five, or a six. It turns out mathematically the expected value is 3.5. Now, as you all know, there's no way you're getting 3.5 if you do it once. But if you keep doing it over and over again, that's what it'll turn out to be on average. That's called the expected value. It's like a long run average. So the expected value if you toss a die is 3.5. And here you can see the logic. If you keep doing this over and over again, posting that die, you'll get as many ones as sixes. That averages out to 3.5. As many twos as fives. So that averages out to 3.5. You'll have as many threes as fours. And that averages out to 3.5. And that's logically why the expected value when you toss a die even once, you're going to say expected value is 3.5.