 or 0.30769. So there's about 0.3 minutes between arrivals. All right, so now we have the question, okay, so that makes sense. Do I want to be talking about this in terms of minutes or do we wanna be talking about it in terms of seconds? Remember, whenever we're looking at time, it gets a little bit confusing because we're base 60 in time. So we gotta be thinking, okay, do I wanna convert this into seconds and be talking about seconds or do we wanna be talking about minutes? In this case, we'll convert it to seconds. So the mean second arrivals, so we're gonna say, all right, I'll take that number times 60, 60 seconds in a minute. That's gonna give us the 18.46. So now we're looking at this in terms of seconds. It takes about 18.42 seconds between arrivals. All right, so then we can look at the X now being for our exponential distribution seconds between arrivals as opposed to the X for our Poisson distribution being the arrivals during a one minute period. So now we can ask questions like, okay, well, what if X was less than or equal to 60, 60 being seconds? So that being one minute, we can use then our exponent dot dist function, which looks a lot like the Poisson dot dist function, X now being the 60 in this case. And then lambda is gonna be one over this 18.46, the mean seconds between arrivals. And then comma the cumulative versus non cumulative. In this case, we want it to be cumulative because I'm adding everything up, the probability between zero up to and including the 60. Okay, so we have that now. We can also plot this out. So if I have this, these are giving us my rows and my rows for the exponential. Remember when I plotted this one out over here, this rows function, I could just make this zero, one, two, and then copy down the rows or I could use a sequence. And if you use a sequence, then that's why these two numbers are here because that gives you a little bit more control to change the numbers that you want in the sequence. So for the exponent dot dist, we used 120. So if I go over here and do a similar thing, now with the exponent dot dist, X now equaling the seconds between arrivals, we can then use our exponent dot dist function, which is gonna be exponent dot dist X is gonna be in this case, the zero, but we copied it all the way down. That's what the hashtag is for. So it's gonna spill. It's gonna be a spill array comma one over this for the lambda. And then the cumulative, this time zero or not cumulative. So it's not cumulative. So then we have the seconds between arrivals. What's the likelihood of zero, 5.42 seconds? What's the likelihood of one second between arrivals, 5.13, two seconds between arrivals, 4.86, three seconds, 4.60 and so on and so forth. So if we were to plot this out then, it would look something like this. So now we have it, this is the characteristic look of an exponential type distribution. And sometimes I feel like the line waiting is a little less intuitive to fully understand. We'll take a look at an example next time. The example of like a radioactive decay declining for some reason that gives me an image, which is another exponential distribution situation, oftentimes in another realm of like science and whatnot, but that usually draws, gives me the picture of this. But the thing to keep in mind is that if you have a Poisson distribution and you flip the question around then you should get generally this exponential, which will give you a characteristic curve that looks like this. Now in future problems, this is just another look of the curve so you can get a fancy curves within Excel. So we'll practice formatting those curves in Excel if you wanna work through the practice problem in Excel. So we'll also take a look at another practice problem to try to get a better intuitive sense as to why and these line waiting situations would this happen because sometimes it doesn't make complete intuitive sense at first, so we'll try to break that out a little bit more in a future presentation and we'll also take a look at it in terms of minutes and instead of in terms of seconds to get a feel for the minutes versus the seconds. The other thing to keep in mind is that if you were to ask a question such as what's the likelihood that you're gonna get, that you have zero up to two, you would think that you can just sum up the percents, but it may not be exact to do that in the exponent.dist situation as easily as you could have done it with a Poisson situation, possibly because of the curve of the exponential distribution possibly needing more complex math in order to do that. So therefore, like calculus, right? So therefore what you would need to do then is use, if you're asking that question, you would need to use the cumulative function. So in other words, if I use the cumulative function to ask the question of the likelihood of zero to three, I may get a different answer than if I just summed up these four numbers possibly due to the curvature of the exponential curve.