0:05

In our previous exercise, we'd looked at using specific mass points to

Â characterize different potential levels of demand.

Â For example, we had said with 30% one level of demand is going to occur,

Â with 50% another level of demand may occur, with 20% another level of demand

Â may occur, with 10% another level of demand might occur.

Â When there are only a few possible outcomes, we can take that approach.

Â But what about when there are many possible outcomes?

Â What if we're trying to predict what sales are going to be?

Â What if we're trying to make predictions as far as, what cost of goods might be?

Â Well, that's where we can use probability distributions to characterize

Â the extent of uncertainty that we have.

Â So that's what we're going to talk about today.

Â 0:50

So as far as the plan is,

Â we're going to talk about how can we use different probability distributions.

Â Some that you may be familiar with, some that might be new to you.

Â But how can we use these distributions to characterize the extent of randomness

Â in the outcomes that we're interested in?

Â So we'll take a look at how we can use these different distributions,

Â where they're going to be appropriate?

Â And of course,

Â what are these Excel commands associated with these different distributions?

Â And then, of course, how do we tie this back to decision making?

Â And we'll tee up another lab exercise where we have

Â another aspect of the profit equation that is uncertain to us.

Â In this case, we're going to use a probability distribution to characterize

Â that uncertainty incorporated into our calculations and

Â use that to make the best business decision.

Â All right, so going back to this framework that we had identified earlier,

Â of course, we're going to build an evaluation model and

Â that's what we're using to make the best possible decision.

Â We just need to make sure that we're incorporating uncertainty in

Â an appropriate way into that evaluation model.

Â 1:53

Right, so, here is one particular example where we have fluctuating levels of

Â demand over time.

Â And what we would like to do is come up with a way of predicting what's

Â my best guess as far as what demand is going to be in the next period.

Â Well, one of the measures that we can use to characterize,

Â to summarize this data is going to be the average.

Â Let me take the average of the historic data that's available to me.

Â And that average might be over a small time period

Â because I want to put more emphasis on the more recent data.

Â But the problem is that's my best guess.

Â It doesn't tell me how much uncertainty I have in that guess.

Â So in this case, if I were to look at it,

Â it seems like demand tends to hover around 30 units.

Â But sometimes it's as high as 60, sometimes it's as low as 5 units.

Â So I might say, well, my best guess is 30, but how likely is that to happen?

Â Also, what we have to take into account is, I might not get 30 units.

Â I might get 31 units, I might get 32 units, I might get 47 units of demand.

Â 2:57

Well, what we want to be able to do is say,

Â here is my best guess, as well as here's how much uncertainty there is around that,

Â so that when we're making our decision, we take that into account.

Â We take into account that our prediction of demand

Â isn't necessarily going to be perfect.

Â In fact, chances are we're not going to predict demand with 100% accuracy.

Â So what's important for us is to recognize how much uncertainty is there?

Â Is it a very tight distribution around a prediction of 30 or

Â is it a prediction of 30 with a lot of uncertainty?

Â So, we're going to talk about ways a little bit later on to say

Â based on factors that we observed, based on time of year,

Â based on marketing activity, we can come up with that prediction for demand.

Â But not only do we predict demand,

Â we also predict how much error we might have in that prediction.

Â And incorporate both of those factors into our decision making.

Â All right, so some terminology that we're going to be using,

Â and this goes back to what we talked about last time,

Â we're going to be thinking about outcomes in terms of random variables.

Â So if I roll a pair of dice,

Â the outcome that I get each time I roll those dice is going to be different.

Â That's going to be my random variable.

Â If I'm looking at sales, every day that I'm observing sales,

Â that's a different possible outcome that I observe.

Â Some days sales are going to be high, some days sales are going to be low, so

Â we might treat sales as a random variable.

Â 4:24

We're going to build a classification for different types of random variables.

Â If they're a fixed number of possible outcomes, as was the case

Â in the exercise that we had gone through last time with our inventory planning,

Â where there were only three levels or let say there are only four levels of demand.

Â We'd refer to that as a discrete random variable where there are a fixed number of

Â possible outcomes.

Â In a lot of cases, there are an infinite number of possible outcomes.

Â Or there could be a granular number of sales outcomes, but

Â it might be a very wide range.

Â Well, if for a particular range if it can take on any value,

Â that's what we're going to refer to as a continuous random variable.

Â And this classification is going to come in handy when we're thinking about

Â what are the appropriate probability distributions for us to be using?

Â When is it appropriate for me, for example, to use a normal distribution?

Â When it is appropriate for me to use a binomial distribution versus a Poisson

Â distribution, or an exponential distribution?

Â If you're familiar with those different distributions, great.

Â If not, we're going to take a look at each of them and

Â talk about when they're appropriate?

Â Why we might prefer one versus the other?

Â A little bit later on.

Â Right, so, again, if we think about this in terms of risk and

Â return, risk is characterizing that extent of uncertainty where we

Â have calculations such as the standard deviation and the variance.

Â The return, that's our expected value or the average.

Â So these formulas that we're using,

Â 6:03

If we look at the Greek letter mu,

Â often use to symbolize our expected value or the average.

Â Well, the way that we would go about calculating that is,

Â if I can enumerate all possible levels of the variable X.

Â If I can list out all of these different levels,

Â what I need to know is how likely am I to observe all of those different levels,

Â and that's what this probability piece is saying.

Â So if I'm rolling the dice, the possible outcomes range from 2,

Â if I roll snake eyes to 12 if I roll two 6s.

Â Well, how likely are each of those to occur?

Â So I'm going to weigh

Â 7:04

How far am I away from that central tendency?

Â And I don't care if I'm above or below it, I just care that how far away I am.

Â That's why we're going to be squaring that term and

Â weighing that difference by the probability of a particular outcome.

Â So, for both of these calculations the expected value.

Â And whether you're looking at the variance or

Â the standard deviation, think of these as weighted averages.

Â We're not just adding up all possible outcomes,

Â dividing by the number of outcomes.

Â We're weighing the average based on how likely a potential outcome

Â is to be observed.

Â And that's going to be important when we get into things like customer analytics,

Â where we say some customers are worth a lot, some are worth a little.

Â But how likely am I to get a customer that's worth a lot versus a customer

Â that's worth a little?

Â We're going to have to take those probabilities into account.

Â 7:58

Right, so this is returning to the example we have worked through

Â in our previous lab.

Â Where we had said there are five specific levels of demand ranging from 100 all

Â the way down to 300.

Â And we have the probabilities associated with each of those levels of demand.

Â Well, if I wanted to calculate what is my expected level of demand

Â using that weighted sum approach.

Â The first term in this equation says I've

Â got a 30% chance of having a demand of 100.

Â I have 150 level of demand with a 20% chance.

Â My chances of getting 200 units of demand is 30%.

Â 15% chance of 250 units, 5% chance of getting 300 units.

Â Adding all of those numbers together,

Â my expected demand is going to be 172.5 units.

Â Now, we can do these calculations by hand, fortunately if you have the data

Â arranged in a tabular format such as this there's an Excel command, sumproduct.

Â Where the two inputs into that sumproduct equation is you're going to highlight

Â the outcome column.

Â So in this case you would highlight the column that has our demand values ranging

Â from 100 to 300.

Â Enter a comma and then highlight the column that corresponds to

Â our probabilities and that's going to give us that 172.5.

Â So that's one way for us to get a, what's my average level of demands?

Â Same approach would be taken if we wanted to calculate the variance or

Â the standard deviation.

Â So we said the average was a 172.5 units.

Â So what we're going to do for calculating the variances?

Â We're going to calculate for

Â each level of demand how far away are we from the average.

Â So 100-172.5,

Â that's how far away this particular observation is from the average.

Â And we're going to square that,

Â 9:54

multiply it by the probability of that observation happening.

Â We do that for each of the different levels of demand that we have.

Â So 300 minus the 172.5 average squared only happens 5% of the time,

Â it gives us a variance of just over 3,600.

Â Taking the square root of that gives us the standard deviation of about 60 units.

Â 10:20

So, using the variance, using the standard deviation, it's a way for

Â us to characterize a degree of dispersion.

Â Just how much dispersion do we have in the data?

Â Now if we have for a particular case the normal distribution,

Â 10:35

the standard deviation has a very simple interpretation for us.

Â If we think about a bell curve,

Â how much of the likelihood is contained within one standard deviation,

Â within two standard deviations, within three standard deviations.

Â What you're going to see today is that the normal distribution,

Â that's going to be the work horse for us.

Â In a lot of cases, using the normal distribution is going to be a appropriate,

Â it's familiar to us and a lot of our intuition is based around that.

Â But we're also going to look at alternative distributions, because that

Â normal distribution isn't always going to be what we're observing in marketing data.

Â 11:12

All right, so if we take a look back at the exercise that we completed

Â in inventory planning which said, our average is 172.5 units of demand.

Â But if we take a look at our expected profit, turns out that if we're only able

Â to order in increments of 25 units, our best bet is to order only a 150 calendars.

Â And if I could order exactly a 173 calendars,

Â I've actually got a lower profit.

Â So, in this case, turns out I'm better off ordering fewer calendars.

Â And so what we're essentially saying is,

Â based on the business context that we're looking at,

Â because when I order calendars, if I don't sell them, I end up losing money.

Â I can return them for some of what it cost me, but

Â I end up losing money when I order too much.

Â Turns out we're better off ordering fewer calendars than our expected level

Â of demand.

Â And so we don't just want to calculate, what's my expected level of demand?

Â We need to take into account the cost structure in the business problems

Â that we're dealing with.

Â So, the approach that we're generally going to take, and

Â we'll do that in the exercise that we work through in this session,

Â is to specify the profit equation.

Â That's our evaluation model.

Â If I knew for example the level of demand that I'm getting,

Â let's write out our profit equation.

Â And then let's characterize the extent of uncertainty that we have using

Â the appropriate distributions.

Â And we're just going to conduct simulations that take into account how

Â likely are different levels of demand.

Â And what we may see is, in this, as we did in this inventory planning exercise,

Â I'm better of ordering fewer calendars under this particular cost structure.

Â There maybe other cases where I maybe better off actually ordering

Â more than the expected level of demand.

Â Or exactly equal to the expected level of demand.

Â So we don't just want to stop with calculating statistics.

Â What we want to do is incorporate those in to that evaluation model.

Â And then use that tool that we've developed

Â to identify what is the appropriate decision for us to make.

Â