 to how long it takes for the next person to arrive. We're gonna try to generate this with a random number generator, which is a little bit more complicated because you can't just use the normal random number generator, and we don't have the Excel functions to give a random number generation similar to what we had with the binome.dist and the Poisson distribution. So we'll make up our own kind of formula here, and which is gonna be this. So it's gonna be equal to negative Lm, which is a natural logarithm. I won't go into that in detail, but note what we're trying to do here is generate the randomness of the numbers that still has a random element to it, but it follows kind of the conditions of what would be present in a natural setting that has a exponential distribution kind of relationship to it. So then we're gonna say one minus the rand, this is the normal random number generator that we've seen in the past for coin flips and whatnot, and then we're dividing that by this mean arrival rate in minutes, the 1.66 on forever. So don't worry too much about that formula, just the concept is we're imagining that we're going out there, we're actually testing this out with our stopwatch, and this is a random number generator that gives us an approximation of that, in what would happen in real life if our line situation followed a Poisson distribution, which would then you would think also follow when you look at the interim times and exponential distribution. So for example, this first one, we had the first customer came in and the enter arrival time was 14, and then the next enter arrival time was 1.16 minutes, and then this is in minutes, by the way, instead of seconds as our prior example, and then the third, it took 2.83 minutes, and then the next arrival 5.81 minutes, 4.15 minutes, 1.1 minutes, 0.15 minutes, 0.07 minutes, and notice the trend that you kind of see when you're looking at this, you had this fairly large one here, and then you had a lot that are pretty low, your five is kind of big, but it's not too large, you've got a lot that are fairly low, six, three, four, seven, and then it jumps up to 10. So now you had an interval between customers of 10, 12 fairly high, and then it goes all the way back down to four, 12, it popped back up four, three, nine, four, 14, jumped up to 14. So you've got a lot that are kind of in the lower range, and then it jumps all the way up to 21, which is a pretty, we haven't seen anything that high for some time, right? And then you've got a lot of lower stuff, and that's what we would generally expect oftentimes in these kind of line waiting situations, and which is why it gives the kind of the character of the curve sloping downward that we've seen with that exponential distribution. So if I took the mean of this data, if I just took the average of this data, we're now getting to 6.49, which is fairly close to six, because we used that six as our kind of interim, that was our mean of our data set that we kind of put into our calculation with this 0.166 over here. So the mean is kind of what we would expect with our randomly number-generated numbers. And so then what I would do in Excel is just copy this whole thing because this cell has this random number generation in it. That means that these cells are always gonna regenerate randomly. So what I'm doing, I would just copy the same data over. These numbers are different because you could see it basically juggled the numbers around, but the idea is that this is still being generated with this function, but now we hard-coded the numbers so they don't change because of that random number generation tool. Okay, so now we can count the frequency of these items, and we can do our standard kind of frequency type of, type of, let's do it this way, where we would say, okay, here's our bends. So these are representing the minutes between arrivals now. So we've got zero, if I look at these numbers, what's the count that we have zero minutes on up to the 40 minutes between the arrival times. And note that you can't really use a count if function to do this because these numbers over here are now not whole numbers. So we have to use the frequency which are gonna give us the bends. So here's our fancy frequency, which we're just gonna take the data array over here, and then we're gonna say the bends, which are over here. That's gonna be our array function that'll give us the frequency, and then it'll put these items into our buckets. So in this case, then we have the number one. So one minute, how many times did we have the one minute? And in our data set on this side, 51 times, two minutes or up to and including, you know, two minutes, we had 30 of those three minutes, 31, four minutes, 19, five minutes, 20, six minutes, 18, seven minutes, 14, eight minutes, 14, nine minutes, 18, 10 minutes. And you can see it starts to go down.