Let's continue our journey on understanding theoretical probability distributions. In this video, we are going to look at normal distribution. Now the learning objectives we have already discussed in the video on uniform distribution. In particular, in this video I'm focusing on normal distribution. Normal distribution is what you might have seen in your high school classes as bell-shaped curve or in your introductory statistics class, wherever you took it. It's one of the most commonly utilized distribution in any type of statistical analysis. That is because in most naturally occurring phenomenon, at least in aggregate, when you take lots of different things occurring at the same time. When you aggregate them, they follow some kind of a bell curve. It makes intuitive sense. For example, the height of a population, Let's say the average height of a population. For example, male in United States is five feet, nine inches. You'll find many more people that are around that height. As you move away from that average value, you will find fewer and fewer people who are too short or too tall. So therefore, we use normal distribution extensively in simulation analysis because while many of the values may not follow exactly normal distribution, they follow a similar pattern. Again, the key characteristics of normal distribution are that there is low probability of occurrence of values as we move away from the mean. And usually we don't define any minimum or maximum as we did in case of uniform distribution. But the probability would be near close to 0 as we move away from the mean. So let's look at a few different normal distributions. So here I'm showing three different normal distributions. Normal distribution with a mean of 0 and a standard deviation of one. Normal distribution in blue, which is mean of five and a standard deviation of two. And then a normal distribution where mean is 10 and a standard deviation of five. The general principles are very similar to uniform distributions. The normal distribution with a mean of 0 and a standard deviation of one is often referred to as a standard normal distribution. Because we can transform any other normal distribution into a standard normal distribution by using a simple formula, which is represented here as Z equals X minus mean of that distribution divided by the standard deviation of that distribution. So if we take any normal distribution and apply this formula. It is transformed into a standard normal distribution of where the mean is 0 and standard deviation is one. So we saw that these different distributions as represented here are of different heights. Again, a similar kind of phenomena that we saw in uniform distribution. For visualization purposes, what I've done here is I have converted all the values into standard normal distributions so that I can show them on the same graph, but minus four, for example, for a distribution that is, has a mean of five and a standard deviation of two refers to actually a number minus 3. For the distribution itself, the transformation brings it to minus 4 under the standard normal distribution. And we can calculate the corresponding values for normal distribution of 10, a mean of 10, and a standard deviation of five. Now the calculation of probability for a value and for example, a number, one or minus one. And below is a little more complicated than because calculating area under the curve is non trivial. But this probability can be easily calculated by most scientific calculators. Many of the statistical books will have these values printed. Or as we'll do, we'll use Excel to calculate this probability. Incidentally, it's 0.1587 or 15 percent approximately for standard normal distribution. Again, it means different things for different distributions. For a standard normal distribution, we are getting a value of minus one or below. For a normal distribution with a mean of five and standard deviation of two, the probability corresponds to a value of minus three or below, and plus five or below for a normal distribution whose mean is 10 and standard deviation of five. Similar to the uniform distribution, we also look at the cumulative distribution function, or we can draw cumulative distribution functions. In case of normal distribution, they look like what are called s-curve or sigmoid functions, We often refer to. Again, drawing them by calculating these values is difficult, so we usually use some software approach or use the values given in tables. So what we're going to do next is we're going to go to Excel and see how these values can be obtained from the software in Excel. And then we will explore exponential distribution.
