Let's look at the last theoretical probability distribution that we wanted to look at in this course. And that's exponential distribution. Again, we have covered the learning objectives earlier. And this particular video, we're going to look at exponential distribution, which is usually used to model the arrival processes in many arrival or service processes in many of the models. So in general, the difference between exponential distribution and the other two distributions that we looked at, uniform and normal, is that the exponential distribution is a skewed distribution. If you think about the uniform and normal distribution and drew a line vertically across the mean. The distribution look the same whether you look to the left of the mean or to the right of the mean. However, in exponential distribution, that's not the case. The second big difference is that in both of the prior distributions, we had to specify two parameters. In the case of uniform distribution, we specified the minimum value and maximum value. And in the case of normal distribution, we had to specify mean and standard deviation. But exponential distribution is based on a single parameter, which is usually called lambda or depicted by lambda. And it reflects the rate of arrival or service or whatever you want to model. The skewed distribution is of a certain type. In this case. In this case, the function starts at a high value and then rapidly declines as far as PDF or the probability distribution is concerned. And the cumulative distribution function then just takes the reverse shape where it rapidly rises and then stabilizes around one. So let's look at a few different exponential distributions. So, as I mentioned, it's often used to model inter-arrival times. for arrival of customers or cars at an intersection or service completion by a server. Single parameter, which we call Lambda. So for example, a lambda of three per minute means that the arrival rate is three individuals per minute. If we look at the shape, as I mentioned earlier, the shape looks like It starts at a high number, so it's more likely that we have smaller inter arrival times or the arrivals will be quicker based on the mean, of course. And then the longer time differences between successive arrivals gradually become less probablistic or less likely. So we see the value of three different functions here. Again, depending upon the arrival rate, you might have, you might start with the lower value and then decline rapidly. Or you can start at a very high value and decline rapidly. As you can see, for lambda equals 1, we start low, and then the decline is therefore lower also. for a higher Lambda values, Which again means that more customers are arriving per unit of time. Therefore, the inter arrival times are going to be smaller and more likely closer together. So again, the interpretation is that it lets us estimate what is the probability of getting a customer in the next 30 seconds. And for example, if you're looking at the first distribution, distribution with lambda equals one, the shaded area under the curve again represents that probability. Again, this is simply calculated by a very simple formula, one minus e to the power minus lambda x. So you can compute it by using a simple scientific calculator. But we again, are probably going to just use Excel to do this because most of our modeling will be contained there. So again, the cumulative distribution function looks just reverse here. So remember, in the case of normal distribution, we had a sigmoid kind of function. In case of uniform distribution, we had a simple line or a linear function. Here we have differently curved function where we start low and rapidly increase let me try to summarize what we have covered in the last few videos in this first lesson for the second week. Essentially we looked at the distributions, probability distributions, and see how we can get the values of interest, probability values of interests from that. So that we can utilize that in our simulation modeling. Essentially probability distributions capture trends, and we try to get a handle on this trend and then try to incorporate it in understanding any phenomenon. Again, while the actual distributions from in a given phenomena in real world event may not follow exactly a specific distribution. But as long as they can approximate it reasonably, we can use these stylized versions of the distributions. In the next video, we're just going to do a lab on exponential distribution in Excel to see how we can do these things in Excel. And then in the next lesson, we're going to take an actual empirical distribution and create an empirical distribution out of it to see how we can define a PDF and CDF of an actually observed data. And use that to model our parameter.
