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Welcome to the last module of An Intuitive Introduction to

Â Probability: Decision Making in an Uncertain World.

Â This last module covers the normal distribution, perhaps the most famous and

Â most important probability distribution in everyday applications.

Â However, before we can really talk about the normal distribution and

Â the famous bell curve, we have to talk about the concept of a continuous

Â random variable and a continuous probability distribution.

Â And that's what this first lecture of this module is all about.

Â Let's dive right straight in with the definition.

Â A continuous random variable X can take on a continuum of possible values.

Â Some people would say an uncountably infinite number of values.

Â 0:52

For example, many variables in every day life are not discrete,

Â they don't jump from value to value, like if you roll a fair die.

Â One, two, three, four, five, six, or in a casino, at a roulette table, zero, one,

Â two, three, four, five and so on.

Â Instead, they take on continuous values.

Â Here are some examples.

Â A time to finish a task or the length if I produce an item,

Â the lengths of the item, the thickness, the width.

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Now you may say, wait a minute.

Â But time, I count in hours, in minutes, in seconds.

Â Yeah, but wait a minute.

Â This is only our human limitation in measuring time.

Â We can think then of hundredths of a second,

Â thousands of seconds, microseconds, nanoseconds.

Â Time doesn't jump, time goes continuously.

Â If we think of it as discrete, it's only because our limitation of measurement.

Â 1:52

So, continuous variables do exist.

Â In addition, believe it or not, sometimes it's easier to think of

Â a variable as being continuous, than instead of having it discrete values.

Â Stock prices are certainly discrete.

Â We go in cents, or here in Switzerland in rappen.

Â Nevertheless, it's sometimes easier to model these random

Â processes with a continuous random variable.

Â So, continuous, that's really a tricky concept.

Â And now I want to spend the rest of this lecture thinking what is

Â continuous variable?

Â What does this really mean?

Â So, let's think of a random variable that can take on any real

Â number between zero and one.

Â And then I can ask a question, and let's start with a little in-class question.

Â What is the probability that this random variable will take on exactly

Â the number zero point, one, two, three, four, five, six, seven, eight, nine?

Â Think about this, what is this probability?

Â Before I give you the answer to this in-class question,

Â I want you to have a look at a little spreadsheet that I prepared.

Â Because Excel has this beautiful random variable function, RAND,

Â that allows us to simulate the random variable on zero, one.

Â Let's do that.

Â So, here now, I prepared a little spreadsheet for

Â you using the random number function in Excel.

Â Here, in every version of Excel, you have this beautiful function RAND().

Â This particular function gives you a random number between zero and one.

Â And so, now look at these numbers.

Â They are all different.

Â Now, what do you think is the chance I get a 0.123456789?

Â Let me try this again.

Â We have these random numbers, I click Enter, we get ten new numbers.

Â Look at this, they change all the time.

Â However, if you want to bet on a particular number, this is hopeless.

Â If you think there's any chance 0.5, 0.55 or from the in-class quiz question,

Â 0.123456789 shows up, it's not going to happen.

Â Probability is zero.

Â As a little aside, for the techies among you, here in Excel,

Â of course, these are not truly random numbers.

Â They're limitations to the computer.

Â They're only so-called pseudorandom numbers.

Â And I only get a limited number of digits here,

Â so technically, we only have finitely many numbers.

Â However, this is meant as a representation of the true continuum.

Â And in that sense, the probability of every number is zero,

Â and we cannot hit any number that I give you ahead of time.

Â So, let's now wrap up this idea,

Â move back to the slides, and continue with continuous random variables.

Â There's really an infinite number of numbers between zero and one, and

Â we cannot count them.

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If you think you can count them, give me the number that comes after zero.

Â If you say it's 0.001, I say no.

Â There's another number that comes first, 0.000001.

Â And even there are many numbers, infinitely many that come before that.

Â Similarly, you can't give me the next number after 0.5 or

Â give me the last number before 0.5.

Â So, we see this continuum doesn't allow us to give positive probabilities,

Â because there's so many numbers, that the total sum of probabilities, in the end,

Â would be larger than one, and we know that cannot be.

Â Think back to the exams we had way back in the beginning of the course.

Â So, the probability of every individual number must be zero,

Â so, or said differently, no individual number can have a positive probability.

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So, what can we still do at this point?

Â Let's think about another in-class question here.

Â What is the probability that this random value that I showed you in the Excel sheet

Â comes up with a number between zero and a quarter?

Â Think about that.

Â [SOUND] [SOUND] So,

Â between zero and a quarter, there are also infinitely many numbers.

Â And if you play with the random number sheet,

Â with those random numbers I showed you, you will notice that in

Â 25% of all cases, you get a number between zero and a quarter.

Â Similarly, between a quarter and a half,

Â 25% of the time you fall into that interval.

Â And a quarter of all numbers are between 0.5 and 0.75, and similarly, 0.75 to one.

Â This looks awfully uniform.

Â And that's the name of this particular distribution that we simulated

Â in the Excel spreadsheet.

Â It's a uniform distribution on the interval 0,1.

Â And there's nothing special on having the width of the interval a quarter.

Â Here, I show you intervals of the lengths of 0.1 between 0 and

Â 0.1, 0.1 and 0.2, all the way to 0.9 and

Â 1, they all show up with 10% probability.

Â 9:11

Any number between zero and one is possible.

Â It has probability zero, but we get a number between zero and one.

Â We don't get negative numbers.

Â We don't get numbers larger than one.

Â Now, look at this rectangle here.

Â A square, we have the whole area is really one.

Â It has a width of 1, it has a height of 1, so the whole area is 1 times 1.

Â Now, here we look at the probability between 0.18 and 0.28.

Â This has a width of 1, and a height of, sorry,

Â a width of 0.1 and a height of 1.

Â So this has an area of 0.1, within the whole area of 1.

Â So, this is now 10% of the whole area.

Â And this is how this green rectangle within the larger

Â box represents the probability, and this is how we do it in general.

Â As a quick aside, for those of you who remember integration from high school,

Â there, we also look at areas underneath a curve.

Â Those are integrals.

Â And if you look at an integral from a point a to the same point a,

Â that integral for any function was always equal to zero.

Â So, here, the representation of probabilities underneath curves

Â corresponds to what you learned in high school about integration.

Â That's just lesson aside for those of you who know this.

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this function needs to be greater equal zero for all elements.

Â We cannot have negative elements.

Â It's either equal to zero or positive.

Â The entire area underneath the curve has to be one, why?

Â Think back to the very beginning, the basic rules of probability.

Â The probability of the sample space, everything that's possible,

Â always must be one.

Â The same thing is true here.

Â That's now, in technical terms, it means the integral of the entire

Â area has to be equal one, that's for the techies among you.

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And then I can use this function now to also calculate the probabilities for

Â fixed ranges, as a probability over an interval,

Â as I showed you in the earlier examples.

Â There now the probability that a continuous random variable falls between

Â a lower number a, and a larger number b, is then just the difference of

Â the cumulative distribution at b, and minus cumulative distribution at a.

Â And so here, I show you the technical

Â calculation that goes along with this very intuitive picture.

Â 13:29

So, to wrap up this very long, but very important lecture,

Â when we talk about the normal distribution, we need to get a good

Â feel for continuity, what it means for a distribution to be continuous.

Â Therefore, we briefly talked about continuous random variables and

Â then looked at the most simple continuous distribution, namely the uniform on 0, 1.

Â And the key takeaway that I need for you to understand

Â is a representation of probabilities as areas underneath a curve.

Â That's a key concept which we will need to calculate probabilities for

Â some cool applications as we go forward.

Â