 to the right, and those are some of the terms that we use to describe the distribution of the data. So intervals between cars at a toll booth or Adam decay. So these are going to be some examples of different types of distributions. We haven't spent as much time on, in prior presentations, we'll spend more time on in current presentations. When we look at these line waiting situations, like waiting in line at a toll booth, for example, then oftentimes there's a pattern that we can see with those types of representations, which is like a Poisson distribution, which we'll talk about shortly, and it has characteristics shapes that we can describe in terms of the characteristics we talked about in the past, in this case skewed to the right, and then we could have exponential distributions we will also talk about as well. So types of data shapes, so the types of data shapes, we could describe our data shapes. Remember if we have our data in the histogram, we could, for example, have a single peaked histogram, which most common values in the center and fewer values as we move away. That's what you might envision more like a bell shaped curve. So we would describe that as having more of the data in the middle with a single peak to it. Symmetric, the data looks the same on both sides of the center. So if it's a symmetric, again, you're probably envisioning like a bell shaped curve with the middle point, and then you have the data somewhat symmetrically on either side of that middle point. But when it is skewed, that's the term we used, right skewed tail on the right of the center, meaning you've got that more data that's going to the right side, and you have that tail that's going out towards the right, and then left skewed tail on the left of the center, the opposite, and you could have a binomial which has two peaks of the data. So instead of having just the data in the middle and then spreading to the side, you could have those two peaks of the data. These are just some terms that we can use to represent the data. And remember when you're looking at different data sets, you could have these, we're trying to, like if you're looking at the landscape here, and that was representing a particular data set, we could try to look at any particular data set and use those general terms to get an idea of what the data set is doing. So now we want to, once we get an idea of being able to kind of describe the histogram with those general terms, we want to be able to see, is there a mathematical description of the data? If we can describe it mathematically with some type of curve or line, that's what's going to give us more predictive power. That's going to be what our focus is more here. So we're going to take a look at some families of distributions now. These are some common families of distributions. One's going to be the uniform distribution. We'll talk a little bit more about each of them in future in a little bit here. You got the Poisson distributions. We've got the exponential distributions and the binomial distributions. So let's take a look at each of those in a little bit more detail and we will do example problems in this section related to some of these families of distributions. So we've got the uniform distribution. This is the easiest one to start thinking about. So in other words, if you're thinking about a set of data, we're trying to say, is this set of data, the histogram that's coming from it, something that I can represent with one of these mathematical formulas, and the first one is a straight line. So that would be a flat line distribution. An example would be rolling a fair die. So in other words, if you roll a dice, you only have one through six that the dice could roll and you would expect then the distribution to be uneven distribution between all the numbers if it was a fair die, which would be an easy function f of x equals c. And if I was to make a histogram of it, it would look like this, right? If I rolled the dice, I think this is representing rolling the dice a thousand times or something like that, pulling out the trustee calculator. So if I rolled one die, you would expect it to be one over six. That's the likelihood 16.66% that it's going to be either a 12345 or six. If I rolled the die 1000 times, then what would you expect to happen that times 1000? You would expect to have about 167 of each number rolled. That would be that would be the what you would expect. Now notice that this is just an approximation, a model of what might happen in the actual world and you can clearly see that because it's impossible for me to roll 166.62 because I can't roll 0.6 of a two, right? That's impossible. So the model is not an exact representation of what could actually happen in the world. But you can see how it gives predictive power of what we would basically kind of expect to happen. You can use that same kind of concept we thought about in the past, which was we're using kind of like a sample. So again, the idea would be if you have the entire population, if you were looking at everything, we weren't in the cave, but we were looking at everything and we could see the actual vision of everything, then you would have that even distribution in this kind of representation. So because we're taking just a snippet, a sample, then it's not, then we're taking an unperfect representation of the world, right? But in any case, we would have just this line, it would just be a line. And so then so so so when we actually roll the dice, it's not going to come out to exactly this line if I roll the die like 1000 times, but this will approximate what what we think should happen. And therefore I can use just the graph of a line to predict what's going to happen. And if I rolled it less than 1000 times, it would have a family of uniform distribution curves or lines, which would be one over six.