Welcome to the second week of this mini course on simulation. This week has three lessons. In the first lesson, we're going to learn about some classical probability distributions that we often use in simulation modeling. We use these distributions when we do not have detailed data available. Then in the second lesson, we are going to use observed data when we have a lot of data available to construct these distributions so that we can use them to simulate. Finally, in the third lesson, we're going to put all of this together by building our first detailed simulation model and capture some insights into a problem that we have already discussed. We're going to start with uniform distribution, which has one of the simplest . So when can we use these theoretical distributions? Well, there are some conditions. We have to know some general shape or central tendencies such as average, either known or estimated. if we have data, we can also formally test whether our data fits a given distribution. If we don't have data, but we have an idea about the central tendencies, then we can use these well-known theoretical distributions. The most common distributions that we use are uniform distribution, which we're going to cover in this particular video. And essentially, uniform distribution is a distribution where everything has equal likelihood of happening. Then we have normal distribution or a bell-shaped distribution, which has a very wide array of applications because it naturally occurs in lot of phenomenon that we observe. For example, heights of people, income of people, and many other naturally occurring phenomenon. Finally, we have exponential distribution. We often use these kinds of models to model the occurrence of the next event. Now obviously these are not the only theoretical distributions, but these are the most common types of distributions that we use. So we'll understand the properties of these distributions and we'll see how we can use them using some Excel labs. In general, a probability distribution curve allows us to judge the likelihood of an event. We normalize These do have the area under a given curve, under a given probability distribution curve to be equal to one. I'm starting with the uniform distribution because these characteristics can perhaps be best described using different uniform distribution. Again, to reemphasize, uniform distributions are where probability of each value occurring is exactly the same. The range of value is described simply by a minimum value and a maximum value. The probability that something below a minimum value can occur or above a maximum value can occur is 0. Or in other words, it's simply cannot happen. We'll look at three different uniform distributions in order to understand what do we mean by distributions and what information can we get out of these distributions? Here we see the three uniform distributions that are talked about on the last slide. So the first distribution is the uniform 0 to 1 distribution. The second one, slightly flatter one, is the uniform distribution from 0 to 2. Notice the range on the x-axis. And then finally, the third distribution is the uniform distribution from 0 to 5. The key thing to notice here is that the area under each box is one. And the way we achieve that is by changing the height of the box. What that represents essentially is because there are fewer values between these two numbers. each of these ranges. For example, 0 to one has fewer numbers than one to two. Essentially the probability of each number occurring is larger. So again, we adjust the height in the case of the uniform distribution of 0 to one, the height is one. In case of uniform distribution from 0 to two, the height is 0.5. And in case of uniform distribution from 0 to five, the height is 0.2. Intuitively, you can say the height kind of represents the individual probability of a number occurring. The probability that we'll get, for example, the most of the time we are interested in a probability such as what is the likelihood of something less than or equal to 0.5 occurring? Now that probabilty can be represented by the area under the curve and can be calculated for each distribution quite easily. For example, for uniform 0 to one, we just multiply the height by the number that we're looking for. For example, 0.5. and that's 0.5 or 50%. Similarly, 4.5 to occur or 0.5 or less to occur. In the distribution 0 to 2 -- uniform distribution from 0 to two, the probability is going to be 0.25. and 0.1 for uniform distribution from 0 to five, we can represent these probabilities that an event where a value of something or less than that, for example, a value of less than or equal to 0.5 can occur by what we call a cumulative distribution function or CDF for short. here we see the CDF for the uniform distribution and these are straight lines. Again, the way we interpret these is that these are probabilities for a number or less. So for example, if we are looking at the uniform distribution of 0 to five and we want the probability of a number three or less occurring, then the probability of that is represented on this graph by this on the y-axis as 0.6. If we want to look at the probability of one occurring with probability distribution from 0 to two, it will be something like 0.5. So again, we can get these probabilities directly from these graphs. And these can directly be obtained from Excel or through a simple calculation as shown on the previous figure. So the actual formula you can use is if the minimum value of the distribution is a, then we can get this value by saying x minus a over b minus a, where b is the maximum value of the distribution. So from 0 to five, It's pretty simple. For uniform 0 to five, let's say. And we want to get a value of three. We will do 3 minus 0 divided by 5 minus 0. And we get three over five, which is equal to 0.6. So we can calculate these values in uniform distribution pretty simply. However, it's not so simple to do it for other distributions. And so we'll use Excel to do those. In the next video, we're going to look at normal distribution. It is a little more complicated to look at in simple terms, but we're going to use Excel to simulate it to get a better understanding of it.