Now that we've seen some examples of probability models in practice, I want to talk about the building blocks of these probabilities, models and in particular we're going to discuss some specific random variables. Both discreet and continuous, those are terms that we've seen a lot when we've been discussing modeling, and so the random variables will come in two forms, discreet and continuous, we'll talk about some very important probability distributions. And, if you recall as I define these objects, a random variable represents a potential outcome of an uncertain event, and the probability distribution. Assigns probabilities to the various potential outcomes, that's the basic idea. So, let's start off by looking at a discreet random variable just to confirm our understanding of the terminology. So anticipate that you're going to roll a die. A single die. And we'll call it a fair die which means each outcome is equally likely. Now, because I haven't rolled the die yet, I don't know what the outcome is going to be. So, that's an example of a random variable and there's various notations, but quite often we'd write the outcome of the random variable as capital X. Now, given it's a six sided die, there are six possible outcomes, and you can see those being illustrated across the first row of the table. And beneath that you can see the probabilities that have been assigned to each of those possible outcomes, and we write, generally, those probabilities as the probability that capital X equals little x, and when you see that first time around it looks a little bit odd. But, what that's trying to say is that capital X is the random variable and little x is the realization of the random variable. So little x can take on the values one, two, three, four, five, or six. And this table displays the probability distribution for rolling a fair die where each outcome is equally likely. So each one is one-sixth. So this is what we mean by a probability model. Now, some facts about probabilities that it's useful to know. The first one, that probabilities have to lie between zero and one inclusive. If anybody ever presents you with a probability greater than one, or less than zero, something has gone horribly wrong. And the other fact about these discrete probabilities is that they have to add up to one. Something has to happen. And so here's an example of a probability distribution. Now that I've shown you a discrete random variable, I want to followup with a continuous random variable. And as an example of a continuous random variable, I'm going to consider the percent change on the S&P 500 stock index. So imagine I asked you, what do you think the S&P 500 is going to close at tomorrow? Well, you don't know the answer to that question, not exactly, and so one might be willing instead to use a probability model to get some assessment of the likelihood of various clothes and prices. And in fact, here I'm not going to look at the price directly, I'm going to look at the percent change sometimes called the return. And the way you would calculate a daily return for a stock or a stock index is to say. What's the price today minus the price yesterday over the price yesterday, that's often term a relative return, and if we multiply that through by 100, we're going to get that on a percentage basis, and that would give us our percent return. Now if I'm talking about the percent return tomorrow, I need to look at tomorrow's price minus today's price over today's price, and that's what's in the formula there, Pt+1-Pt over Pt. So, that's my object of interest, the percent change, and technically that quantity can take a value between mine is 100%, that would be a bit of a disaster, where everything was lost and infinity, I mean that's a little bit technical, but potentially you could get any value between there. Clearly, some feel more likely than others, typically, the returns on the market vary between plus or minus 1% each day, something of that order. Now, when we want to calculate probabilities of continuous random variables, it's a little bit different. We look at what's called the probability density function. I'm going to show you one of these on the next slide. So here's a potential probability distribution of the S&P 500 daily percent changes. And what that will give you is a probability model for the daily percent change. So, notice here that we have a complete curve. And each of the values on the x axis, the percent change axis, is a potential outcome. I haven't drawn this out to plus or minus infinity because if that doesn't really make sense as a very unlikely outcome, so I've captured the majority of potential outcomes here, and the way that you would calculate probabilities from such a graph is by looking at the area underneath the graph. So for this continuous random variables the probability that are associated with areas under this graphs. And if I wanted for example, to ask the question, what's the probability that the S&P 500 falls by more than half a percent, it means a percent change of minus 0.5. Then, the way I would do that is take this graph, I would identify the value minus 0.5 on the x axis, and I said more force by more than minus 0.5%. So that means area to the left of, and so the area under this graph would give you the probability. So in summary, the probabilities associated with the continuous random variables come from calculating errors. Now, in practice, you don't have to calculate these errors, you're going to use software to calculate the area for you. And so in Excel, and sheets, they're going to be built in functions that will calculate these probabilities, these areas on the curves. But the important thing to realize is that, given the model, the probability model really being the shape of this distribution here then given that model we're going to be able to calculate various probabilities. So those are our two sorts of random variables, the discrete and the continuous random variable. Given that we've got a probability distribution, then one of the things we like to be able to do is summarize it in some fashion rather than just showing people the shape. Or the numbers if it's a discrete probability distribution, what we might like to do is summarize it in some fashion, and there are some summaries that are frequently used. And one of the most basic summary is to say where is the center of that distribution? And the measure that we're going to talk about that captures the idea of centrality is known as the mean. And the mean, which I'm writing with the Greek letter mu here, is a measure of centrality. To get a sense of what the mean is doing for you, imagine taking the picture, which is actually a normal. Probability model and cutting out the shape of the graph making it, stamp it out in a piece of metal, then pick it up and try and balance it on your finger. So if you find the balancing point of that distribution you have found the mean, that's what it does. So the mean is a measure of centrality in this particular example here the mean is sitting there at zero right in the middle of the distribution, and that's because it's what's called a symmetric distribution. If you put a mirror in the middle of the distribution, you see the same, whether you look at the distribution or what's happening in the mirror. So, that's the first measure, the first summary, centrality, mu. And after looking at the middle of the distribution, the other feature that we like to capture is the spread of the distribution. Now there are two ways that we measure the spread. One is through what is called the variance. And we write that with the great letters sigma squared. And then, there's the variance very close cousin called the standard deviation, which is just the square root of the variance which we write as sigma. And what this measures are spread I'm telling you is how spread out the distribution is, is it all clumped up in the middle or is there a lot of spread associated with it? And these two numbers are not the only summaries of a distribution, centrality and spread, but they're certainly the fundamental ones. And we will often use these two summaries to provide a characterization of a random variable, so for example, after having run a Monte Carlo simulation, you will have generated one of these distributions with the output from that Monte Carlo simulation. And rather than presenting someone with the whole graph it's useful to present someone with a whole graph, but it's also useful to present people with numerical summaries. And the two most likely numerical summaries you'd want to be able to present would be the mean mu and the standard deviation sigma. Again, the way that those will be calculated is through software.