Original subtitles

Instructor: Hello again.

When we started this section of the course

we mentioned how some events have infinitely

many consecutive outcomes.

We call such distributions continuous

and they differ vastly from discreet ones.

For starters, their sample space is infinite.

Therefore we cannot record the

frequency of each distinct value.

Thus we can no longer represent

these distributions with a table.

What we can do is represent them with a graph,

more precisely the graph of the

probability density function, or PDF, for short.

We denote it as F of Y where Y is an element

of the sample space.

As the name suggests, the function depicts

the associated probability for every possible value, Y.

Since it expresses probability

the value it associates with an element

of the sample space would be greater than or equal to zero.

Great, the graphs for continuous distributions

slightly resemble the ones for discreet distributions.

However, there are more elements

in the sample space so there are more bars on the graph.

Furthermore, the more bars, the narrower each one must be.

This results in a smooth curve

that goes along the top of these bars.

We call this the probability distribution curve

since it shows the likelihood of each outcome.

Now onto some further differences

between distinct and continuous.

Imagine we use the favored overall formula

to calculate probabilities for such variables.

Since the sample space is infinite, the likelihood

of each individual one would be extremely small.

If we assume the numerator stays constant, algebra dictates

that the greater the denominator becomes

the closer the fraction is to zero.

For reference one third is closer to zero than a half

and a quarter is closer to zero than either of them.

Since the denominator

of the favored overall formula

would be so big, it is commonly accepted

that such probabilities are extremely insignificant.

In fact, we assume their likelihood

of occurring to be essentially zero.

Thus, it is accepted

that the probability for any individual value

from a continuous distribution to be equal to zero.

This assumption is crucial in understanding

why the likelihood of an event being strictly greater

than X is equal to the likelihood of the event being greater

than or equal to X for some value X within the sample space.

For example, the probability

of a college student running a mile

in under six minutes is

the same as them running it for at most six minutes.

That is because we consider the likelihood

of finishing in exactly six minutes to be zero.

That wasn't too complicated, was it?

So far we've been using

the term probability function to refer

to the probability density function of a distribution.

All the graphs we explored

for discrete distributions were depicting their PDFs.

Now we need to introduce

the cumulative distribution function

or CDF for short.

Since it is cumulative

this function encompasses everything up to a certain value.

We denote the CDF as capital F of Y

for any continuous random variable Y.

As the name suggests, it represents probability

of the random variable being lower than

or equal to a specific value.

Since no value could be lower

than or equal to negative infinity,

the CDF value for negative infinity would equal zero.

Similarly, since any value would be lower

than plus infinity, we would get a one

if we plug plus infinity into the distribution function.

Discreet distributions also have CDFs

but they're used far less frequently.

That is because we can always add

up the PDF values associated

with the individual probabilities we are interested in.

Good job folks,

the CDF is especially useful when we want

to estimate the probability of some interval.

Graphically the area under the density curve would represent

the chance of getting a value within that interval.

We find this area by computing the integral

of the density curve over the interval from A to B.

For those of you who do not know how to calculate integrals,

you can use some free online software like WolframAlpha.com

if you understand probability correctly, determining

and calculating these integrals should feel very intuitive.

All right.

Notice how the cumulative probability is simply

the probability of the interval

from negative infinity to Y.

For those who know calculus, this suggests

that the CDF for a specific value Y is equal

to the integral of the density function

over the interval from minus infinity to Y.

This gives us a way to obtain the CDF from the PDF.

The opposite of integration is derivation.

So to attain a PDF from a CDF, we would have

to find its first derivative, in more technical terms,

the PDF for any element of the sample space Y

equals the first derivative of the CDF

with respect to Y.

Okay oftentimes, when dealing with continuous variables

we are only given their probability density functions.

To understand what its graph looks like, we.

-: Should should be able

to compute the expected value and variance for any PDF.

Let's start with expected values.

The probability of each individual element Y is zero.

Therefore, we cannot apply

the summation formula we used for discrete outcomes.

When dealing with continuous distributions,

the expected value is an integral.

More specifically, it is an integral of the product

of any element Y and its associated PDF value

over the interval from negative infinity

to positive infinity.

Right, now let us quickly discuss the variance.

Luckily for us, we can still apply the same

variance formula we used earlier for discrete distributions.

Namely, the variance is equal

to the expected value of the squared variable minus

the expected value of the variable squared.

Marvelous work.

We now know the main characteristics

of any continuous distribution

so we can begin exploring specific types.

In the next lecture, we will introduce

the normal distribution and its main features.

Thanks for watching.