 Statistics and Excel. Exponential distribution in seconds. Roller coaster line example. Got data? Let's get stuck into it with statistics and Excel. You're not required to, but if you have access to OneNote, we're in the icon left hand side. OneNote presentation 1576. Exponential distribution in seconds. Roller coaster line example tab. We're also uploading transcripts to OneNote so you can go into the view tab, use the immersive reader tool, change the language if you so choose, be able to either read or listen to the trans. First, a word from our sponsor. Yeah, actually we're sponsoring ourselves on this one because apparently the merchandisers, they don't want to be seen with us, but that's okay whatever because our merchandise is better than their stupid stuff anyways. Like this CPA thinking cap, for example. CPA thinking, CAP, you see what we did with like with the letters, and this CPA thinking cap is not just for CPAs either. Anyone can and should have at least one possibly multiple CPA thinking caps. Why? Because based on our scientific survey of five people, all of whom directly profit from the sale of these CPA thinking caps, wearing this CPA thinking cap without a doubt, according to the survey, increases accounting productivity tenfold. Yeah, at least. Yeah, apparently the hat actually channels like accounting energy from the quantum field ether directly into your head, allowing you to navigate spreadsheets faster. It's kind of like how in like the matrix when Neo learns kung fu, or at least that's what the scientific survey is saying, so get one because the scientific survey participants could really use some extra cash. If you would like a commercial free experience, consider subscribing to our website at accountinginstruction.com or accountinginstruction.thinkific.com. Script in multiple different languages using the timestamps to tie in to the video presentations. One note desktop version here and prior presentations, we've been thinking about how we can represent different data sets, both numerically using calculations such as the average or mean, the quartiles, the median, and pictorially using the box and whiskers and histogram. The histogram being the primary tool we visualize when thinking about the spread of data. And we can describe the spread of data on a histogram using terms such as it's skewed to the left, it's skewed to the right. We're now looking at functions or formulas that have a smooth curve or a line related to them, which oftentimes can approximate actual data sets in the real world. And if we can approximate our data sets with a line, that would be great because it'll typically give us more predictive power over whatever it is we're looking into. In prior presentations, we looked at families of these types of formulas, these curves, including the uniform distribution, the Poisson distribution, the binomial distribution. Now we're looking at exponential distribution. Now the exponential distribution often has a relationship to the Poisson distribution. So oftentimes we're going to kind of flip the question around when thinking about a Poisson distribution to the question that we'll be thinking about with an exponential distribution. So oftentimes with a Poisson distribution, just to set the groundwork, we're talking about in business scenarios, line waiting situations where we ask questions such as, what's the likelihood that so many people will be arriving within some interval of time, like one minute or one second? For example, we also had a problem not over time, but over distance, such as how many pot holes would show up in a mile of road. But we're going to now look at our time example. So if we think first about like the Poisson, and then we'll kind of convert that to the exponential for a line waiting situation for like a roller coaster ride, for example, x is going to be the arrivals during one minute. And we're going to say that the mean is going to be 3.25. So the 3.25 is the mean arrivals within the timeframe distance of one minute. So this is going to be the Poisson distribution, not the exponential. This is just the starting point. If I was to graph this out, x is going to be equal to the arrivals during one minute. So what's the likelihood that zero people arrive in one minute? We're going to say that that's going to be the 3.88 according to our Poisson dot dist function, which is going to be taking the x, which is going to be this is we have a range here, a spill that we're taking. That's why the hashtag is there comma the mean 3.25. And then the cumulative, it's not cumulative. That's why it's going to be a zero because we're looking at just the zero. We did that all the way down. What's the likelihood that one person shows up in the one minute time frame, the 12.6, what's the likelihood that two people show up the 12.48. If I wanted to know the likelihood of having zero to three people show up within the one minute time, I can then say is 3.88 plus 12.6 plus 20.48 and so on. So we looked at that in a similar presentation in our prior practice problems. And we can also look at the the curve would look something like this. So here's our Poisson curve. So now we're going to say, okay, well, now we want to ask a different question and say the minutes between arrivals. So now we're getting to our exponential distribution. So now we're saying, all right, well, if there's going to be a mean of 3.25 people arriving within a one minute time period, what's the what's going to be the mean minutes between arrivals. So to see how we kind of flipped it on its head, we're looking at the minutes between arrivals. Well, that means we can take the the the 3.25 divided by one hold on other way around one divided by one minute divided by the 3.25. And that gives us about our point 30769. So there's about 0.3 minutes between arrivals. Alright, so now we have the question. Okay, so that that makes sense. Do I want to be talking about this in terms of minutes? Or do we want to 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 got to be thinking, okay, I do I want to do I want to convert this into seconds and be talking about seconds? Or do we want to be talking about minutes in this case, we'll convert it to seconds. So the mean second arrivals. So we're going to say, alright, I'll take that number times 60, 60 seconds in a minute, that's going to give us the 18.46. So now we're looking at this in terms of seconds, it takes about 18.42 seconds between arrivals. Alright, 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're going to 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 going to be one over this 18.46, the mean, 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, this is, these are giving us my rows and my rows for the exponential. Remember when I plotted this one out over here, this this rows function, I could just make this 012, 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 going to be exponent dot dist x is going to be in this case, the zero, but we copied it all the way down. That's what the hashtag is for. So it's going to spill. It's going to be a spill array comma one over over this for the lambda. And then 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, you know, 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 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 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 fancy curves within Excel. So we'll practice formatting those curves in Excel if you want to work through the practice problem in Excel. So so we'll and 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 because it doesn't sometimes it doesn't make complete intuitive sense, you know, at first. So we'll try to 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 going to get that you have zero up to two, you would think that you can you can just sum up the the percents. But I don't believe it may not be exact to do that in the exponent dot 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 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.