So, let's start off by looking at a discreet random variable just to

confirm our understanding of the terminology.

So anticipate that you're going to roll a die.

A single die.

And we'll call it a fair die which means each outcome is equally likely.

Now, because I haven't rolled the die yet,

I don't know what the outcome is going to be.

So, that's an example of a random variable and there's various notations, but

quite often we'd write the outcome of the random variable as capital X.

Now, given it's a six sided die, there are six possible outcomes, and

you can see those being illustrated across the first row of the table.

And beneath that you can see the probabilities that have been assigned to

each of those possible outcomes, and we write, generally,

those probabilities as the probability that capital X equals little x, and

when you see that first time around it looks a little bit odd.

But, what that's trying to say is that capital X is the random variable and

little x is the realization of the random variable.

So little x can take on the values one, two, three, four, five, or six.

And this table displays the probability distribution for

rolling a fair die where each outcome is equally likely.

So each one is one-sixth.

So this is what we mean by a probability model.

Now, some facts about probabilities that it's useful to know.

The first one, that probabilities have to lie between zero and one inclusive.

If anybody ever presents you with a probability greater than one, or

less than zero, something has gone horribly wrong.

And the other fact about these discrete probabilities is that they have to

add up to one.

Something has to happen.

And so here's an example of a probability distribution.

Now that I've shown you a discrete random variable,

I want to followup with a continuous random variable.

And as an example of a continuous random variable,

I'm going to consider the percent change on the S&P 500 stock index.

So imagine I asked you, what do you think the S&P 500 is going to close at tomorrow?

Well, you don't know the answer to that question, not exactly, and so

one might be willing instead to use a probability model to get some assessment

of the likelihood of various clothes and prices.

And in fact, here I'm not going to look at the price directly,

I'm going to look at the percent change sometimes called the return.

And the way you would calculate a daily return for a stock or

a stock index is to say.

What's the price today minus the price yesterday over the price yesterday,

that's often term a relative return, and if we multiply that through by 100, we're

going to get that on a percentage basis, and that would give us our percent return.

Now if I'm talking about the percent return tomorrow,

I need to look at tomorrow's price minus today's price over today's price, and

that's what's in the formula there, Pt+1-Pt over Pt.

So, that's my object of interest, the percent change, and

technically that quantity can take a value between mine is 100%, that would

be a bit of a disaster, where everything was lost and infinity, I mean that's

a little bit technical, but potentially you could get any value between there.

Clearly, some feel more likely than others, typically, the returns on

the market vary between plus or minus 1% each day, something of that order.

Now, when we want to calculate probabilities of continuous random

variables, it's a little bit different.

We look at what's called the probability density function.

I'm going to show you one of these on the next slide.

So here's a potential probability

distribution of the S&P 500 daily percent changes.

And what that will give you is a probability model for

the daily percent change.

So, notice here that we have a complete curve.

And each of the values on the x axis,

the percent change axis, is a potential outcome.

I haven't drawn this out to plus or minus infinity because if that doesn't really

make sense as a very unlikely outcome, so I've captured the majority of potential

outcomes here, and the way that you would calculate probabilities from such

a graph is by looking at the area underneath the graph.

So for this continuous random variables the probability that are associated with

areas under this graphs.

And if I wanted for example, to ask the question, what's the probability that

the S&P 500 falls by more than half a percent,

it means a percent change of minus 0.5.

Then, the way I would do that is take this graph,

I would identify the value minus 0.5 on the x axis, and

I said more force by more than minus 0.5%.

So that means area to the left of, and so

the area under this graph would give you the probability.

So in summary, the probabilities associated with the continuous

random variables come from calculating errors.

Now, in practice, you don't have to calculate these errors,

you're going to use software to calculate the area for you.

And so in Excel, and sheets, they're going to be built in functions

that will calculate these probabilities, these areas on the curves.

But the important thing to realize is that, given the model,

the probability model really being the shape of this distribution here then

given that model we're going to be able to calculate various probabilities.