 as we get up to those higher numbers. So there was a lot of them, you could see that are on the lower side. So the time between arrivals, we tend to have a bunch that are on the shorter side when things are following this exponential distribution. And then we've got a few that take a lot longer in the intrams. And that is how you can kind of imagine what's happening with our curve. So then we're gonna say, okay, I can also represent this in terms of a percent of the total. So these frequency bends, if I add them up, should add up to the number of counts that we did over here, the number of customers that we looked at and saw the intram time, which was 300. So that looks correct. And so I can divide each of these then by the total of 300. So 51 over 300, whoops, hold on a second. 51 over 300 gives us the point, one seven or 17%. So there's gonna, so we can represent this as a percent then as well, which is what it's gonna be represented as when we do the actual exponential distribution. And so that's showing that calculation. Okay, so then I could do it, I can do it this way. X equals the arrivals during one minute and let's this time use our actual exponent dot dist. So now I'm gonna do the same thing, not using our randomly generated numbers, which represent us actually going out there with a stopwatch, but now we're just gonna do the smooth curve using our exponent dot dist, where I'm just gonna take the X here, we're gonna take the lambda, and then we're gonna take the cumulative, it's not gonna be cumulative, so we put a zero. So now we're gonna plot this out with our actual curve, which is similar. Notice it's giving us the percentages, right? Because when I use this curve, I'm not gonna get an actual frequency because we're looking at the percentages. So then I'd have to, if I looked at this one, what's the likelihood that we have the one minute? And then if I did it 300 times, you would think the 300 times the point 1411 would be the actual frequency of it. So that's why you need the percent that we have so we can compare over there. And so this is what we get when we get the smooth curve or the curve generated from our function, right? And you can compare these out. So if this is the one, this is versus one, two, and two, three, and three, four, and four, five, and five. And so you could see there's somewhat similar. And so if I was to plot this out, this is the enter arrival times from our actual data set, plotting this out in a histogram, which looks like this. And you can see it kind of, it's approximating the shape that we would expect. It's not perfect, of course, because we didn't generate, we only generated 300 numbers. Here it is with another type of graph. And then if I looked at it in comparison to the actual curve, which is the blue curve in this case. So the blue curve is a nice smooth curve compared to the random generated curve. We can see that it approximates what we would expect from the exponential distribution. So, and so the general idea with these line waiting situations, like why does that happen? And you could see why it kind of happens here is because you've got these, the times are often short, the intervals are often short, but then you have some of those intervals that are the long intervals, right? And that's what's given that characteristic type of shape, which often happens in these line waiting situations. So if you were in a, so if you saw the Poisson distribution in a line waiting situation, then again, oftentimes you would think that if you took the exponential, the time between, that it would follow, this kind of exponential characteristic shape as well.