- 00:00
[MUSIC - MOBY, "REPEATER"][Business Mathematics][Module 9: Probability Part 5: Expected Values, Means,Variances, and Standard Deviations]

- 00:10
RICHARD WATERMAN: So the probability examplesthat we've seen have been driven by [Richard Waterman, AdjunctProfessor of Statistics] a probability model.And the events associated with that model being presentedin a table and the associated probabilities of those eventsbeing placed in the table.

- 00:30
RICHARD WATERMAN [continued]: And then the calculations boil downto summing the probabilities of the simple events.Now imagine though that you've gota big table, a big probability table.It's kind of hard to look at thatand get a good sense of what's going on.And very frequently, we like to summarizemore complicated entities.So if we think of the probability table

- 00:50
RICHARD WATERMAN [continued]: as a complicated thing, I'd love to beable to summarize some key features of that probabilitytable.And we do have a way of summarizingthose probabilities, and it's through the idea of expectedvalue of a random variable.And if we simply take the expected value

- 01:13
RICHARD WATERMAN [continued]: of the random variable itself, then that'swhat's called the mean of the distribution.And we, in terms of notation, often writethat as the Greek letter mu.And so mu is known as the expected valueof the random variable, or more colloquially, as the mean.And if I was being very technically correct,

- 01:36
RICHARD WATERMAN [continued]: I'd call it the population mean-- mu.So mu tells you the expected value of the random variable.So going back to the hurricanes--what's the expected number of hurricaneswe're going to have next year?That's the question that I'm addressing now.So you can see that's a way of summarizingthat whole probability table.

- 01:58
RICHARD WATERMAN [continued]: So the way that we work out one of these expected values--well let me tell you what it is, and then I'llsort through this notation here.These expected values are just weighted averages.And if we were to denote the possible valuesthat the random variable can take by-- I'm writing themas a little x1, little x2, all the way out to little x

- 02:20
RICHARD WATERMAN [continued]: and there's a capital N. So thoseare all the possibilities that the random variable can take.So, for example, if it were a die that we were looking at,then those values, the little x1, little x2,up to a little xN would be 1, 2, 3, 4, 5, and 6,because those are the possible values that it can take.So I write down the list of the possible valuesthat the random variable can take,

- 02:41
RICHARD WATERMAN [continued]: then I write down the probabilities of eachof those events.So, in other words, I'm just writing down the probabilitytable here.The outcomes, x, the probabilities, p.Then the expected value of x that wewrite as a big E of X E for expectation,and we define that as mu.

- 03:01
RICHARD WATERMAN [continued]: It's simply defined as, and I'm nowlooking at the sum at the bottom of the slide, little x1 timesthe probability that you take on the valuex1 plus little x2 times the probability that you takeon the value of little x2, all the way through to little xNtimes the probability that you take on the value, xN.And if you look at that formula, what you're seeing there is

- 03:27
RICHARD WATERMAN [continued]: the sum of the possible values, the sum of the x's.So it's Ike an average in that sense.But the weights associated with each possible outcomearen't just the probability that that outcome occurs.And so when I look at mu there, the population mean,what I'm seeing is a weighted average

- 03:47
RICHARD WATERMAN [continued]: of the possible outcomes where the weights I've justgiven by the probability of each particular outcome.So this expectation is just a weighted averagewhere the weights are the probabilities of the event.And that's a nice, one-number summary of the distribution.What do we expect to happen?

- 04:09
RICHARD WATERMAN [continued]: Now that we have this idea of expected value defined,let's just calculate one of these thingsand see how it works.So I'm introducing here an example.Imagine that you're running a consumer electronics show.It happens somewhere in the country,during the winter where you can get snow happening.

- 04:30
RICHARD WATERMAN [continued]: Well, if you have a consumer show,and it snows really badly, then potentially nobody shows up,and that's pretty disastrous if you're running that event.So it is, in fact, possible to take outinsurance policies against weather-related problems.And this example is looking at what the benefits of taking out

- 04:54
RICHARD WATERMAN [continued]: that insurance policy might be.So this probability model says that the probabilityof snow on the day of the show is 0.05, so a 1 in 20chance of a snow day.And we are thinking of the world as one of two events happening.It's either snowing or it's not.

- 05:16
RICHARD WATERMAN [continued]: And therefore, no snow has a 95% probability of occurrence.So that's sort of definitional.And then according to this model, if it snows,the show has no revenue.And if it does not snow, then the showis going to make $1 million in revenue.What's the expected revenue of the show?

- 05:40
RICHARD WATERMAN [continued]: So in order to calculate this, I need to write downthe probability table again.And my random variable is the revenue,if the show is going to happen this year.It hasn't happened yet.I don't know what the revenue is going to be.It's a random variable.But if it snows, according to this model,then the revenue is zero.If there's no snow, then the revenue is $1 million.

- 06:01
RICHARD WATERMAN [continued]: The probabilities of those two eventsare 0.05 5 and 0.95 respectively.So I've got the probability table written down.I can now go out and calculate my expectation.The expected revenue, we write that with a letter,mu, the Greek letter mu.And that is, just-- before we do the calculation-- just

- 06:24
RICHARD WATERMAN [continued]: remember intuitively what it is, it'sa weighted sum of the possible revenue values wherethe weights correspond to the probabilityof the event occurring.And so what I get is zero revenue with probability 0.05.That's the snow event.

- 06:45
RICHARD WATERMAN [continued]: And $1 million in revenue with probability 0.95.And so I take those two components now and sum themand I end up with $950,000.So according to this model, my expected revenueis going to be $950,000.

- 07:06
RICHARD WATERMAN [continued]: Now the insurance company comes alongand sells a policy for $50,000.And that policy will pay the event organizers halfa million, or $500,000 if it snows.If it doesn't snow, then the organizersget nothing from the insurance company.

- 07:27
RICHARD WATERMAN [continued]: So what's the expected net revenueif the organizers buy the policy?So rather than working out the raw revenue,as we were on the previous slide,we're going to talk about the net revenue under the scenariothat the policy has been purchased.And again, we're in one of two weather situations.

- 07:47
RICHARD WATERMAN [continued]: It's either going to snow or it's not going to snow.But now, because we've purchased the policy,our revenues are going to change.Don't forget, when we look at the net revenue,to subtract the $50,000 that has beenspent by the show organizers on purchasing the policy itself.

- 08:09
RICHARD WATERMAN [continued]: So if it snows, then the net revenueassociated with the show is going to be $450,000.You get $500,000 back from the insurance company,but you had to pay them $50,000 in the first place.Now, the probability of that event occurringis still 0.05 because your purchase of the policy

- 08:31
RICHARD WATERMAN [continued]: does not impact the weather.So those probabilities aren't changing,it's just the revenues, the net revenue that is changing.Now under the no snow scenario, well, you spent $50,000on the policy, but you got nothing backbecause there was no snow.You didn't need to use the policy.So your net revenue is now $950,000.

- 08:53
RICHARD WATERMAN [continued]: Now if you look at the expected net revenue now,it's the same calculation as before,just slightly different numbers.You're going to find a weighted averageof the net revenues, weighted by the probabilitiesof their occurrence.And going through the math I get 450,000 times 0.05

- 09:15
RICHARD WATERMAN [continued]: plus 950,000 times 0.95, and that turns out to be $925,000.So there's the expected net revenueon purchasing the policy.Now notice that the expected net revenueis less than the expected revenue

- 09:37
RICHARD WATERMAN [continued]: if you didn't buy the policy.If you didn't buy the policy, it was $950,000.Now it's $925,000.So you might say, well, why would I ever buy the policy?Because at least as it's presented here,I've got a lower expected net revenue.Well, the reason you might choose to buy the policy

- 10:01
RICHARD WATERMAN [continued]: is what I'm going to talk about next.And that is because by buying the policy,you can reduce the variability or the risk in your revenue.So this question-- why bother buying a policyif your expected revenue is lower after having boughtthe policy-- is actually answered

- 10:23
RICHARD WATERMAN [continued]: through another summary of the probability distribution.And that summary or attribute or featureis called the variance of the distributionof the random variable.And I'm going to define variance on the next slide.So I said the mean, the expected value of the random variable,

- 10:45
RICHARD WATERMAN [continued]: describes the center of a probability distribution,to some extent, the middle where we expect to be.But the variance describes another featureof the distribution, and that featureis how spread out the random variable is potentially.Now variance is a bit more complicated then.

- 11:09
RICHARD WATERMAN [continued]: So it corresponds to the spread of the random variable,but it is absolutely fundamental in financebecause it's synonymous with risk.When a finance person talks about risk,it really is equivalent to someonewho deals with probability talking about variance.And so variance is definitely an important idea.

- 11:30
RICHARD WATERMAN [continued]: In terms of notation, the varianceis often given the notation sigma squared.So the sigma is the Greek s, little s-- sigma squared.And it's the spread of the random variable about its mean.

- 11:51
RICHARD WATERMAN [continued]: And variance gets defined by the expected value.Now not of x, but of how far is x away from the mean?So we're looking to see the expected departurefrom the mean, but we have to do that on a squared level.So the variance is the expected squared deviation

- 12:15
RICHARD WATERMAN [continued]: from the mean.So that's how we understand spread--the expected squared deviation from the mean.And again, we write that as sigma squared.The calculation for all of these expectations is similar.To find the expected value of anything,you just find the probability weighted sum of that anything.

- 12:36
RICHARD WATERMAN [continued]: And so here we're looking at variation from the mean mu,so the variance would be x1 minus mu squaredtimes the probability that the random variable takeson the value x1, plus x2 minus musquared times the probability the random variable takeson the value x2, all the way out to the last component, xNminus mu squared times the probability of xN.

- 12:60
RICHARD WATERMAN [continued]: So look at that formula.The p's, the probabilities, are the weights.And so we're just finding this weighted sumor weighted average of the underlying componentswhere the components here are the squared standard deviationsfrom the mean.So that's how we measure the spread of the random variable.And we call it the variance.We give it the notation sigma squared.

- 13:24
RICHARD WATERMAN [continued]: So let's do one of these calculations now.Again, we'll do it on the insurance companyfor the variability of the revenuesunder scenario one, where we don'tget the policy, and scenario two,where we do take out the policy.So let's do the variance calculation now.It's a little bit more complicated,but given that this probability model is quite straightforward,

- 13:46
RICHARD WATERMAN [continued]: it's not too bad.And so first of all we'll find a variance under the scenariothat the policy is not taken out.And so under this first scenario,remember what the mean mu was?It was 950,000.So I'm going to write down a table that will allowme to do the calculations.

- 14:07
RICHARD WATERMAN [continued]: And so the weather event is going to be snow or no snow.If I don't take out the policy, then the revenueis either going to be zero or $1 million.And now, in the row of the table,Revs minus mu all squared, I'm working outthe individual components, the deviationof the random variable from the mean squared.

- 14:31
RICHARD WATERMAN [continued]: So I work out under the snow scenario, the revenue is 0.How far is 0 from the mean?Well at 0 minus 950,000 and square it.And then under the no snow scenario,then my revenue is $1 million.And how far away is that from 950,000?Well I do the subtraction and then square it.

- 14:51
RICHARD WATERMAN [continued]: And so Revs minus mu squared are the componentsthat I'm interested in.And on the last row of the table are the probabilities, 0.05and 0.95 again, showing us the weightsthat we are going to multiply the outcomes by,the probability weights.So now if I work out that expectation,it's simply by definition 950,000 squared times 0.05

- 15:17
RICHARD WATERMAN [continued]: plus 50,000 squared times 0.95.And you're just saying, where did that 50,000 come from?It comes from doing 1 million minus 950,000in the table above.So those are the components.Go to the calculator, work them out,and it comes out to be a rather large number.I think that's 47.5 billion.

- 15:38
RICHARD WATERMAN [continued]: So that's the variance associated with the revenuewhen I don't take out a policy.So let's have a look at that variance whenI do take out a policy.So I'm about to replicate this calculation,but for the scenario where we take out the policy.

- 15:59
RICHARD WATERMAN [continued]: So if we want to take out the policy,then the table changes a little bit.Remember, we then have net revenuesof either 450,000 under the snow scenarioor $950,000 under the no snow scenario.When we take out the policy, the expected value,or mu decreased, remember?It went down to $925,000.

- 16:22
RICHARD WATERMAN [continued]: So I have to work out the deviations about thatmean in this particular case.And so the components are going to be under the snowy scenario,$450,000, my Net Revs, minus the mean $925,000 all squared.And under the no snow scenario, $950,000 minus $925,000, all

- 16:45
RICHARD WATERMAN [continued]: squared.So those are the components.I'm going to multiply them through by the probabilities,0.05 and 0.95, to get that the variance under this scenario,sigma squared for this scenario, if you plug into the calculatorand work it out, you will get about $11.875 billion

- 17:06
RICHARD WATERMAN [continued]: under this scenario.And what you can see if you compare the two variances,and again, these are the variance or the spreadof the risk ultimately associated with this revenue,is that if we take out the policy,that that variance has fallen by a factor of about four.

- 17:28
RICHARD WATERMAN [continued]: And so given that variance is synonymous with risk,the whole point of the policy is that wereduce the variability associated with the revenuesthat we have.And so even though the expected revenuesdrop on taking out the policy, the variability

- 17:48
RICHARD WATERMAN [continued]: drops as well, which is good.The variability of the revenues drops,which is good because it means that there's less risk.And so that's, to some extent, what insurance is all about.You pay a little for the policy, but you reduce your riskby having done so.And these calculations show this to you very, very directly.

- 18:12
RICHARD WATERMAN [continued]: You can understand why it's so usefulto be able to take out an insurance policy.Let's say you were running the show.You've got to pay, for example, the people who work for you.You need some money.Let's say you need some cash flow.And so if you don't take out the policy and it snows,you get zero revenue.So how are you going to pay everyone who works for you?That's a very bad place to be.

- 18:33
RICHARD WATERMAN [continued]: And so you might well be willing to lose some of your profit,or lose some of your revenues, your expected revenues,by taking out the policy.And the benefit of having done sois that you are going to reduce the risk or the variabilityassociated with that revenue stream.

- 18:56
RICHARD WATERMAN [continued]: So even if it snows, you're probablygoing to be able to pay your employees.So that's why we deal with insurance.So the main ideas are captured by these two featuresof the distribution, the mean that wewrite as mu, and the variance, that we write as sigma squared.Now there's one more summary that I

- 19:17
RICHARD WATERMAN [continued]: need to introduce to finish off this section.I will do so now.So the variance, we saw in our calculatorthere was a kind of a big number, somewhat unwieldy.And part of the reason for that isthat variance is defined in terms of squared deviationsfrom the mean.

- 19:37
RICHARD WATERMAN [continued]: And we know that squares grow quickly.Another problem with that squaringis the units of the variance.And so if, for example, x is measured in dollars,as it was in the insurance example,then the variance of x is going to be measured in dollarssquared.And that's a pretty hard quantity to interpret.

- 20:00
RICHARD WATERMAN [continued]: Someone pays you $5 squared-- not meaningful.And so it's hard to interpret.That leads us to define a very close cousin of the variance,and it's called the standard deviation.The standard deviation is definedas the square root of the variance,

- 20:20
RICHARD WATERMAN [continued]: and so if we're writing the variance as sigmasquared, then its square root just must be sigma,and that's the standard deviation.And one of the benefits of using the standard deviationto describe the spread of a probability distributionis that it has units, that is that it

- 20:41
RICHARD WATERMAN [continued]: has the same units of x.So if x is measured in dollars, then standard deviationis measured in dollars.We also see in next time's class whenwe talk about statistics some interpretationsof that standard deviation.But those will wait until next time.For now we'll just note that the standard deviation has

- 21:02
RICHARD WATERMAN [continued]: the benefit of being measured on the same scaleas the random variable.So let's calculate those standard deviations nowfor the two scenarios-- one, wherewe don't take out the policy, two, wherewe do take out the policy.All that we have to do is grab the calculatorand take some square roots.And under the no policy scenario,

- 21:23
RICHARD WATERMAN [continued]: the standard deviation of our revenues is about $218,000.You can see we're back to the sort of unitsthat the problem was set up in.In terms of getting revenue of $1 million,our standard deviation is the same orderof magnitude $218,000.That's under the no policy scenario.

- 21:45
RICHARD WATERMAN [continued]: And then if I were to take out the policy,the weather-related policy, then my standard deviationof the revenues, my net revenues,drops to about $109,000.So it is half.Not surprisingly, if the variance dropby a factor of four, then the standard deviationis going to drop by about a factor of two,

- 22:06
RICHARD WATERMAN [continued]: square root of four is two.So you can see how that standard deviation is decreasingwhen you take out the policy and you are reducing your risk.One word that people throw around a lotis out of volatility.People talk about volatility of returns,

- 22:27
RICHARD WATERMAN [continued]: volatility of the market.Volatility is just another way of saying standard deviation.If a finance person wants to measurethe volatility of a sequence of stock returns,they're just going to find the standard deviation.And so they say a lot of these probability conceptsare actually core to finance.

- 22:48
RICHARD WATERMAN [continued]: Not surprisingly, because financeobviously deals with uncertain quantities.[Music: Repeater by Moby, courtesy of mobygratis.com][Business Mathematics][Richard Waterman]

### Video Info

**Series Name:** Business Statistics

**Episode:** 5

**Publisher:** Wharton

**Publication Year:** 2014

**Video Type:**Tutorial

**Methods:** Probability, Random variables, Standard deviations

**Keywords:** insurance and risk; mathematical computing; mathematical concepts; weight

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Richard Waterman discusses expected value, mean, variance, and standard deviation. He introduces these concepts for use in probability models. Using a probability model, Waterman calculates the risk and benefits of an insurance policy.