0:36

Here is sort of a technical, mathematical explanation for that.

Â I just want to briefly mention this, it's not really important as we move on.

Â But let's briefly talk about the central limit theorem.

Â This is a very cool math theorem which says the following.

Â If you add many effects that are a little random but

Â have roughly the same variance, the same volatilities, the same variation.

Â Then the compound process of these many little processes,

Â in the end has the normal distribution.

Â 1:14

And the amazing thing is that this very technical, very difficult,

Â don't look at the proof, it's really nasty.

Â It has a nasty mathematical proof, it's ugly, it's very, very difficult.

Â Nevertheless, this very technical result is

Â extremely relevant in everyday applications, why?

Â Because we have processes where lots of little things come together.

Â If you think about markets, in particular,

Â large markets where we have many suppliers and many buyers.

Â Many sellers and many buyers,

Â there we have lots of little opinions of what should be a price.

Â And as a result, there we see the normal distribution in action.

Â Similarly, in production processes where

Â we have lots of little things going on in the production of an item.

Â 2:26

Then, other important applications are in the field of statistics.

Â When we there talk about the probability distributions of sample means,

Â of sample proportions.

Â Guess what?

Â The normal distribution shows up as the samples get large.

Â In business, there's in the area of operations management,

Â something called quality management, the so-called Six Sigma, and

Â there the normal distribution is used.

Â In finance, people have a very important risk concept of value at risk,

Â there the normal distribution has been used a lot.

Â Recently, it has been criticized there, for some various reasons.

Â Nevertheless, probability distributions close or

Â similar to the normal distributions, are used there to this day.

Â So, now the normal distribution, this bell curve, the bell curve

Â is an example of a density function that we saw in the previous lecture.

Â This particular density function is bell shaped,

Â it's symmetric around the mean, which is exactly in the middle.

Â And the spread is determined by the standard deviation, sigma.

Â 4:02

Here just to scare a living daylight out of you,

Â I'll show you the density function.

Â It has the number pi, it's a famous geometry constant 3.14 in it.

Â And the Euler number, 2.71828 and infinitely many more digits,

Â named after the Swiss Russian mathematician Euler.

Â And again, here you see the picture.

Â Now, let's look at this graph.

Â What happens to the distribution, to the bell shape if you change mu or sigma?

Â If you first look at the red, green, and

Â blue graph here, they all have the same mu.

Â That means they're all centered at the same number, they all have the peak,

Â the maximum, at the same number.

Â 5:12

Now, the last curve, the purple one has a smaller mu,

Â that means it moves to the left.

Â It's now centered at a smaller value.

Â So, the key idea, mu gives us point where it's centered,

Â and smaller mu moves the curve to my left,

Â larger mu moves the curve to my right.

Â Sigma, the smaller sigma, the more narrow the curve gets and the higher it is.

Â 5:46

Again, the key concept from the previous lectures.

Â The bell curve itself doesn't give us probabilities,

Â it's the areas underneath the bell curve, those are the probabilities that we need.

Â And remember, we can calculate those with a cumulative distribution function.

Â Because it's so important, let me repeat it one more time.

Â For continuous distributions such as a normal distribution,

Â probabilities are areas underneath the curve.

Â And we calculate those with the cumulative distribution function.

Â Now we have a problem.

Â There is no close form formula, there is no simple formula for

Â the cumulative distribution function, capital F,

Â of the normal distribution, we can only approximate these areas.

Â In the old days, before we had modern computers and

Â cool software, people had to do table lookups.

Â Some people either by hand, or the supercomputers at a time

Â calculated these tables and they were printed in probability books.

Â Sadly, nowadays still some people think that's the way to go.

Â No, it's not, it's very old-fashioned, it's so the last millennium.

Â So my advice, rip out those pages out of your probability textbook and

Â roast marshmallows on them, they are useless.

Â Nowadays we want to use software packages.

Â If you use some fancy mathematical software packages,

Â it does have the normal distribution.

Â But you don't need even that.

Â Excel can help you to calculate any normal probability that

Â you may care about, and that's what we do here.

Â The function we are going to use is a NORM.DIST function in Excel,

Â here it's written down, it has four arguments that we need to enter.

Â Let me show you right away an example of this function in action.

Â Here we have a normal distribution with a mean of 63, a standard deviation of 5.

Â And I want to know, what's the probability that this random

Â variable takes on a value less than 65, or larger than 65.

Â What do I need to type in?

Â NORM.DIST, the value I care about right now is x65.

Â 8:16

Second number is the mean, 63 in this case.

Â The third number is the standard deviation, sigma, 5.

Â And finally, the last entry is TRUE, or 1.

Â Remember back to the binomial distribution?

Â TRUE or 1 always means we're working with the cumulative distribution function.

Â And that's the only thing that makes sense here.

Â So don't even think about it, just type in TRUE.

Â And we see, there it goes, 0.6554 is the probability

Â that this random variable will take on a value of less than 65.

Â Using the complement rule, 1- this probability gives us

Â the probability of the right tail, 0.3446.

Â So you see, we don't need complicated tables and look for

Â numbers in a complicated table.

Â We just type a one-line command in Excel and we get the probability we care about.

Â 9:12

So, let me wrap up this lecture.

Â We talked about the famous bell curve, which is the density function or

Â the graph of the density function of the normal distribution.

Â And calculations of those probabilities from the normal

Â distribution is really easy.

Â We can just use the NORM.DIST function in Excel and

Â your Excel does have that function.

Â So in the next lecture we now look at examples, and

Â start having fun with this function and calculate normal probability.

Â So please come back for more fun with the normal distribution.

Â Thank you.

Â