Welcome back to an intuitive introduction to probability, decision making in an uncertain world. Now in this lecture, I want to show you an example of an application of means, variances, and standard deviations. One are where random variables are heavily used is in the area of finance. When finance practitioners and academics talk about Financial Models essentially they are talking about random variables. And they then use these random variables or these economic models to forecast because they need forecast to handle economic or financial risks and that's what this is all about. Again, they have complicated probability distributions that are to difficult to understand so they like to summarize these numbers in these summary measures that we have seen before. Let's start with an example here. Here, we have a model of an investment fund. We have six different return scenarios. In the next quarter we may loose 15% of our money, or we may lose 7.5% of our money, bla, bla, bla. Or we make all the way to 14% of our money. In an excel spreadsheet that comes with the course you can see all these numbers and calculated cumulative probabilities. These probabilities that we see here on the returns they are actually empirical probabilities. Thinking back to module one of this course probabilities we are using here, probability concept number two. Based on historical values you think these are reasonable probabilities. With 2% you loose 15%, with 5% probability you loose 7.5%, and you believe that there's a 20% chance that you make a lot of money of 14% return. And then, in the last column we see the cumulative probability. Now, many investors who want to invest They don't want to look at six numbers and these probabilities. They are interested in sum metrics. And now the first metric that people can look at is exactly our expected value. So here we have the expected value. Let's think of this fund that some one wants to invest in as a random variable. Here, we have the simplified random variable where we have six possible values. We can calculate the expected return. Remember how this goes? You take the first value, multiply it by it's probability take the second value, multiply it with it's probability do that for all values and then sum up all these products. Et voilà! Here we get a number of 4.95. So this would now say that, calling to this model the expected return, that is our unit here the expected return of this investment find is 4.95%. But now, danger! There's no sign of risk. It says: "Oh, the average value is 4.95%." It doesn't mean I will make 4.95%. No! Remember the interpretation, this would only be true if this model is correct over long, long, long investment horizon, many, many quotas of investment. But we're only thinking about one quota now. So, here we have again the problem of information loss in the expected value. We now need to model the volatility. As they say in finance or we use a term variance and standard deviation. So, let's calculate the variants. Remember the variants, all the deviations squared times the probability of these deviations. Here, I quickly calculated for you, and we get a variance of this return of 43.8975, and now comes the killer percentage squared. Why do we have this funny unit? Here, look at the formula we have, -15%-4.95% that difference of 19.95% gets squared and now we get these squared units but they don't mean anything to anyone therefore, we now need an extra step we take the square root of the variance to obtain the standard deviation, 6.63% and this is now our measure of the volatility some people go further for the riskiness of this investment. They say that the standard deviation of this funds return is about 6.63% because we took the square root we went back to regular units the percent, and as I said this standard deviation is a popular measure in economics and science of the concept of the volatility. How volatile is this investment. So here, now you saw a nice real-world business application on expected value, variance and standard deviation with an application to an investment fund.