Class 1 MGMT 370 with Lonnie Turpin McNeese State Operations Management (Part 1)

this is a review for management 370 with Dr Turpin I'm going to be doing a review of some of the practice problems we did today and I will attach the PDF of them below so you can follow with me um I'll try to kind of read through them but it might be a lot of text so if you maybe pull them up side by side that could be a good idea all right so for number one we have an exponential um distribution problem and it's asking us it says the cumulative distribution function evaluated at aal5 minutes in between arrivals is shown in the graph using the appropriate formula verify that the probability of a being less than5 equal 39 so what is actually happening here what is on the graph that I have not drawn yet because I want to show you guys is the concept that at5 minutes which is 30 seconds there will be a probability of point there will be a 39% chance that someone will walk into this Bank let's just use a bank for example so the way to draw this is by going on our X AIS our time and saying 30 seconds so5 and we're asked to verify that it is 39 so what can be confusing about the graph is that you're not going to go to the X AIS for the answer like you might normally you're not going to go to right here and it'll say 39 it's not like that instead it is calculating this entire region so we are finding this region and that will give us the probability and um the reason it's like this is because as time goes on the probability that someone will walk into the bank increases and so if we were just taking it at a minute in time like at 1 minute it would be less but since we're counting the um this is called a probability density function because it measures the density or how much is filled of the graph and so to to verify that this region is in fact a probability of 39 we going to use a formula and that formula is this the probability of a being less than5 = 1 - e Lambda T negative Lambda negative Lambda T and so um the E will be on your calculator and we have been told here that the Lambda equals 1 and we've been told that we'll be given that so no need to sweat that let's plug in the numbers we're going to do equal sign1 and then our T is our time so that'll be 0.5 so all you have to do is plug in those numbers and make sure that when you do it on your calculator you plug it in all at once so it doesn't subtract the E first or else you'll get the wrong answer but basically this should give you 39 and um this is our final answer we just want to use the formula to verify that that is correct so good job we're already done with the first one all right the second one says what is the marginal probability that the gap between arrivals exceeds 2.5 minutes so for this one we are trying to find the exact same thing but instead of if it's um instead of our less than formula we're going to use our greater than formula so we can go 2.25 this time we're trying to find this region so the formula that we need to use is [Music] this and this is I filled it in because it's the exact same formula as before without the one minus and then like this was our formula before and now our formula is just this oh wait take away that so the only difference is that um you don't do one minus if it's greater than and if you work this out on your calculator it should give you1 there we go all right next we're um asked to find the joint probability so what is the probability that a customer will come into the bank in between 1 and 2 minutes so this time we're trying to find this probability and how do we do that cuz the formula for that would be very long here's what we're going to do we're going to kind of do it in two parts so um for the first part we're we're going to find just imagine this is the same graph we're going to find um the probability that it's between one and two I mean between 0 and 2 minutes and then all we have to do is subtract the probability that's in between 0 and 1 minute if that makes sense so to find this we're going to go using that formula because it's the less than so we do have the one minus and that's going to be um awesome light work then I'll do the next one in [Music] brown so if you you see like if we take this and subtract this it'll give us this that's just how we do it we're going to use that same formula but this time we're going to substitute a one here for this one and that's going to give us 63 so now all we have to do there we go so the probability of someone arriving um in between 1 and 2 minutes is 23 excuse me all right so the next question is what is the mean and standard deviation of the variable a what do you notice about these two and so this is an interesting one I'll I'll I'll just work the problem and then we can see oh what's what is it what is it going to be so the formula for the mean pretty simple don't don't fully process what this means just yet um but this is what the mean formula is then standard deviation what is that and this is how it's denoted and then often times this is how it's denoted I think in our class we're going to do it like this this is just some ways that people write it and so standard deviation equals and so what is what is the same or different about these well you kind of have to already know the answer before so I'll just work it and you can see basically um if we basically the stand deviation there's another formula that goes like this and so what we can do is plug this [Music] in and um the square root can just go on top and the bottom so it'll go and since it's a square root of one it'll just be one and since it's Square rooting a square it'll cancel out so this is what our standard deviation looks like and now do you notice anything good job our standard deviation and our mean are the same so this only happens in an exponential um this distribution because it's a continuous random variable so it is like when it says continuous it's at the same rate and it's random um yeah I don't think we don't need to go too much into it I think we'll get into it more in the semester but it's just good to know that mean and standard standard deviation will be the same all right I think I'll do one more and then it we go into section two so I'll do that in a in the next video this one says calculate the marginal probability that a is greater than e of a so a our random variable is greater than the mean now think about this for every variable a that follows an exponential distrib of the form a x a and it has a squiggle lamb Lambda prove the expression probability of a is greater than the mean then equals e to the 1 and so that's a lot to hear but if you go in the PDF um in the link it'll look really simple I'm just going to write our equation and then solve it for us sorry and so so it's wanting us to solve this and um both these we just found are one was given the Lambda and then the negative one um I mean the negative one yeah we did on the first one so we have um one was our Lambda and then we we have the negative so it becomes one and then the other one comes from um let's see this problem already it gives us this e to 1 I believe Okay cool so all you have to do is plug this into your calculator and E will equal 37 and so what we're finding what we basically just found was on that graph we found this area and so where these two numbers came from the one we have a negative from the equation already and then we are told the Lambda was a one and since we're at 1 minute that's for our time and then e was just on our calculator how you can press you can press like um to enable the blue buttons I think it's second then the division sign that'll give you your e but it this was just a super simple formula and so this is this probability region is 37 and final thing I'll say thank you for watching I hope this has been helpful it's pretty cool because the probabilities have inverse relationship so the probability of someone coming into the bank um from time 0 to 1 minute will be um 63 so if you don't know one or the other you can just flip it um but this was day one part one I will try to do part two tomorrow see you