Welcome back to an intuitive introduction to probability, decision making in an uncertain world. Now we are ready to learn about the normal distribution, the perhaps most famous probability distribution. Many people have seen, at some point in their life, the famous bell curve. Now, the first question you may have, why is this bell curve, the bell shape, so important? Because it shows up in many real world applications. Here is sort of a technical, mathematical explanation for that. I just want to briefly mention this, it's not really important as we move on. But let's briefly talk about the central limit theorem. This is a very cool math theorem which says the following. If you add many effects that are a little random but have roughly the same variance, the same volatilities, the same variation. Then the compound process of these many little processes, in the end has the normal distribution. And the amazing thing is that this very technical, very difficult, don't look at the proof, it's really nasty. It has a nasty mathematical proof, it's ugly, it's very, very difficult. Nevertheless, this very technical result is extremely relevant in everyday applications, why? Because we have processes where lots of little things come together. If you think about markets, in particular, large markets where we have many suppliers and many buyers. Many sellers and many buyers, there we have lots of little opinions of what should be a price. And as a result, there we see the normal distribution in action. Similarly, in production processes where we have lots of little things going on in the production of an item. Where we have various variables from our surroundings like air pressure, temperature and so on. They compound to something that's often very well approximated by the normal distribution. Then, other important applications are in the field of statistics. When we there talk about the probability distributions of sample means, of sample proportions. Guess what? The normal distribution shows up as the samples get large. In business, there's in the area of operations management, something called quality management, the so-called Six Sigma, and there the normal distribution is used. In finance, people have a very important risk concept of value at risk, there the normal distribution has been used a lot. Recently, it has been criticized there, for some various reasons. Nevertheless, probability distributions close or similar to the normal distributions, are used there to this day. So, now the normal distribution, this bell curve, the bell curve is an example of a density function that we saw in the previous lecture. This particular density function is bell shaped, it's symmetric around the mean, which is exactly in the middle. And the spread is determined by the standard deviation, sigma. This distribution only depends on two parameters, the mean, which is given by the mu, and the standard deviation, given by a number sigma. If you and I agree on mu and sigma, we indeed have the same normal distribution. Here just to scare a living daylight out of you, I'll show you the density function. It has the number pi, it's a famous geometry constant 3.14 in it. And the Euler number, 2.71828 and infinitely many more digits, named after the Swiss Russian mathematician Euler. And again, here you see the picture. Now, let's look at this graph. What happens to the distribution, to the bell shape if you change mu or sigma? If you first look at the red, green, and blue graph here, they all have the same mu. That means they're all centered at the same number, they all have the peak, the maximum, at the same number. They are distinguished themselves with the level of sigma. Notice the larger sigma gets, the larger the standard deviation, the larger the spread. That means the curve, the maximum, goes down, and the curve goes wide out. Now, the last curve, the purple one has a smaller mu, that means it moves to the left. It's now centered at a smaller value. So, the key idea, mu gives us point where it's centered, and smaller mu moves the curve to my left, larger mu moves the curve to my right. Sigma, the smaller sigma, the more narrow the curve gets and the higher it is. Again, the key concept from the previous lectures. The bell curve itself doesn't give us probabilities, it's the areas underneath the bell curve, those are the probabilities that we need. And remember, we can calculate those with a cumulative distribution function. Because it's so important, let me repeat it one more time. For continuous distributions such as a normal distribution, probabilities are areas underneath the curve. And we calculate those with the cumulative distribution function. Now we have a problem. There is no close form formula, there is no simple formula for the cumulative distribution function, capital F, of the normal distribution, we can only approximate these areas. In the old days, before we had modern computers and cool software, people had to do table lookups. Some people either by hand, or the supercomputers at a time calculated these tables and they were printed in probability books. Sadly, nowadays still some people think that's the way to go. No, it's not, it's very old-fashioned, it's so the last millennium. So my advice, rip out those pages out of your probability textbook and roast marshmallows on them, they are useless. Nowadays we want to use software packages. If you use some fancy mathematical software packages, it does have the normal distribution. But you don't need even that. Excel can help you to calculate any normal probability that you may care about, and that's what we do here. The function we are going to use is a NORM.DIST function in Excel, here it's written down, it has four arguments that we need to enter. Let me show you right away an example of this function in action. Here we have a normal distribution with a mean of 63, a standard deviation of 5. And I want to know, what's the probability that this random variable takes on a value less than 65, or larger than 65. What do I need to type in? NORM.DIST, the value I care about right now is x65. Second number is the mean, 63 in this case. The third number is the standard deviation, sigma, 5. And finally, the last entry is TRUE, or 1. Remember back to the binomial distribution? TRUE or 1 always means we're working with the cumulative distribution function. And that's the only thing that makes sense here. So don't even think about it, just type in TRUE. And we see, there it goes, 0.6554 is the probability that this random variable will take on a value of less than 65. Using the complement rule, 1- this probability gives us the probability of the right tail, 0.3446. So you see, we don't need complicated tables and look for numbers in a complicated table. We just type a one-line command in Excel and we get the probability we care about. So, let me wrap up this lecture. We talked about the famous bell curve, which is the density function or the graph of the density function of the normal distribution. And calculations of those probabilities from the normal distribution is really easy. We can just use the NORM.DIST function in Excel and your Excel does have that function. So in the next lecture we now look at examples, and start having fun with this function and calculate normal probability. So please come back for more fun with the normal distribution. Thank you.