The following content is provided under a Creative
Commons license.
Your support will help MIT OpenCourseWare continue to
offer high quality educational resources for free.
To make a donation or view additional materials from
hundreds of MIT courses, visit MIT OpenCourseWare at
ocw.mit.edu.
PROFESSOR: So today's agenda is to say a few more things
about continuous random variables.
Mainly we're going to talk a little bit about inference.
This is a topic that we're going to revisit at the end of
the semester.
But there's a few things that we can
already say at this point.
And then the new topic for today is the subject of
derived distributions.
Basically if you know the distribution of one random
variable, and you have a function of that random
variable, how to find a
distribution for that function.
And it's a fairly mechanical skill, but that's an important
one, so we're going to go through it.
So let's see where we stand.
Here is the big picture.
That's all we have done so far.
We have talked about discrete random variables, which we
described by probability mass function.
So if we have multiple random variables, we describe them
with the a joint mass function.
And then we define conditional probabilities, or conditional
PMFs, and the three are related according to this
formula, which is, you can think of it either as the
definition of conditional probability.
Or as the multiplication rule, the probability of two things
happening is the product of the probabilities of the first
thing happening, and then the second happening, given that
the first has happened.
There's another relation between this, which is the
probability of x occurring, is the sum of the different
probabilities of the different ways that x may occur, which
is in conjunction with different values of y.
And there's an analog of all that in the continuous world,
where all you do is to replace p's by f's, and replace sums
by integrals.
So the formulas all look the same.
The interpretations are a little more subtle, so the f's
are not probabilities, they're probability densities.
So they're probabilities per unit length, or in the case of
joint PDf's, these are probabilities per unit area.
So they're densities of some sort.
Probably the more subtle concept to understand what it
really is the conditional density.
In some sense, it's simple.
It's just the density of X in a world where you have been
told the value of the random variable Y. It's a function
that has two arguments, but the best way to think about it
is to say that we fixed y.
We're told the value of the random variable Y, and we look
at it as a function of x.
So as a function of x, the denominator is a constant, and
it just looks like the joint density.
when we keep y fixed.
So it's really a function of one argument, just the
argument x.
And it has the same shape as the joint's density when you
take that slice of it.
So conditional PDFs are just slices of joint PDFs.
There's a bunch of concepts, expectations, variances,
cumulative distribution functions that apply equally
well for to both universes of discrete or
continuous random variables.
So why is probability useful?
Probability is useful because, among other things, we use it
to make sense of the world around us.
We use it to make inferences about things that we do not
see directly.
And this is done in a very simple manner
using the base rule.
We've already seen some of that, and now we're going to
revisit it with a bunch of different variations.
And the variations come because sometimes our random
variable are discrete, sometimes they're continuous,
or we can have a combination of the two.
So the big picture is that there's some unknown random
variable out of there, and we know the distribution that's
random variable.
And in the discrete case, it's going to be given by PMF.
In the continuous case, it's given a PDF.
Then we have some phenomenon, some noisy phenomenon or some
measuring device, and that measuring device produces
observable random variables Y.
We don't know what x is, but we have some beliefs about how
X is distributed.
We observe the random variable Y. We need a
model of this box.
And the model of that box is going to be either a PMF, for
the random variable Y. And that model tells us, if the
true state of the world is X, how do we expect to Y to be
distributed?
That's for the case where Y is this discrete.
If Y is a continuous, you might instead have a density
for Y, or something of that form.
So in either case, this should be a function
that's known to us.
This is our model of the measuring device.
And now having observed y, we want to make
inferences about x.
What does it mean to make inferences?
Well the most complete answer in the inference problem is to
tell me the probability distribution
of the unknown quantity.
But when I say the probability distribution, I
don't mean this one.
I mean the probability distribution that takes into
account the measurements that you got.
So the output of an inference problem is to come up with the
distribution of X, the unknown quantity, given what we have
already observed.
And in the discrete case, it would be an object like that.
If X is continuous, it would be an object of this kind.
OK, so we're given conditional probabilities of this type,
and we want to get conditional distributions of the opposite
type where the order of the
conditioning is being reversed.
So the starting point is always a formula
such as this one.
The probability of x happening, and then y
happening given that x happens.
This is the probability that a particular x and y happen
simultaneously.
But this is also equal to the probability that y happens,
and then that x happens, given that y has happened.
And you take this expression and send one term to the
denominator of the other side, and this gives us the base
rule for the discrete case.
Which is this one that you have already seen, and you
have played with it.
So this is what the formula looks like in
the discrete case.
And the typical example where both random variables are
discrete is the one we discussed some time ago.
X is, let's say, a binary variable, or whether an
airplane is present up there or not.
Y is a discrete measurement, for example, whether our radar
beeped or it didn't beep.
And we make inferences and calculate the probability that
the plane is there, or the probability that the plane is
not there, given the measurement that we have made.
And of course X and Y do not need to be just binary.
They could be more general discrete random variables.
So how does the story change in the continuous case?
First, what's a possible application of
the continuous case?
Well, think of X as being some signal that takes values over
a continuous range.
Let's say X is the current through a resistor.
And then you have some measuring device that measures
currents, but that device is noisy, it gets hit, let's say
for example, by Gaussian noise.
And the Y that you observe is a noisy version of X. But your
instruments are analog, so you measure things on
a continuous scale.
What are you going to do in that case?
Well the inference problem, the output of the inference
problem, is going to be the conditional distribution of X.
What do you think your current is based on a particular value
of Y that you have observed?
So the output of our inference problem is, given the specific
value of Y, to calculate this entire function as a function
of x, and then go and plot it.
How do we calculate it?
You go through the same calculation as in the discrete
case, except that all of the x's gets replaced by p's.
In the continuous case, it's equally true that the joint's
density is the product of the marginal density with the
conditional density.
So the formula is still valid with just a
little change of notation.
So we end up with the same formula here, except that we
replace x's with p's.
So all of these functions are known to us.
We have formulas for them.
We fix a specific value of y, we plug it in, so we're left
with a function of x.
And that gives us the posterior distribution.
Actually there's also a denominator term that's not
necessarily given to us, but we can always calculate it if
we have the marginal of X, and we have the model for
measuring device.
Then we can always find the marginal distribution of Y. So
this quantity, that number, is in general a known one, as
well, and doesn't give us any problems.
So to complicate things a little bit, we can also look
into situations where our two random variables are of
different kinds.
For example, one random variable could be discrete,
and the other it might be continuous.
And there's two versions.
Here one version is when X is discrete, but Y is continuous.
What's an example of this?
Well suppose that I send a single bit of information so
my X is 0 or 1.
And what I measure is Y, which is X plus, let's
say, Gaussian noise.
This is the standard example that shows up in any textbook
on communication, or signal processing.
You send a single bit, but what you observe is a noisy
version of that bit.
You start with a model of your x's.
These would be your prior probabilities.
For example, you might be believe that either 0 or 1 are
equally likely, in which case your PMF gives equal weight to
two possible values.
And then we need a model of our measuring device.
This is one specific model.
The general model would have a shape such as follows.
Y has a distribution, its density.
And that density, however, depends on the value of X.
So when x is 0, we might get a density of this kind.
And when x is 1, we might get the density
of a different kind.
So these are the conditional densities of y in a universe
that's specified by a particular value of x.
And then we go ahead and do our inference.
OK, what's the right formula for doing this inference?
We need a formula that's sort of an analog of this one, but
applies to the case where we have two random variables of
different kinds.
So let me just redo this calculation here.
Except that I'm not going to have a probability of taking
specific values.
It will have to be something a little different.
So here's how it goes.
Let's look at the probability that X takes a specific value
that makes sense in the discrete case, but for the
continuous random variable, let's look at the probability
that it takes values in some little interval.
And now this probability of two things happening, I'm
going to write it as a product.
And I'm going to write this as a product in
two different ways.
So one way is to say that this is the probability that X
takes that value and then given that X takes that value,
the probability that Y falls inside that interval.
So this is our usual multiplication rule for
multiplying probabilities, but I can use the multiplication
rule also in a different way.
It's the probability that Y falls in
the range of interest.
And then the probability that X takes the value of interest
given that Y satisfies the first condition.
So this is something that's definitely true.
We're just using the multiplication rule.
And now let's translate it into PMF is PDF notation.
So the entry up there is the PMF of X evaluated at x.
The second entry, what is it?
Well probabilities of little intervals are
given to us by densities.
But we are in the conditional universe where X takes on a
particular value.
So it's going to be the density of Y given the value
of X times delta.
So probabilities of little intervals are given by the
density times the length of the little interval, but
because we're working in the conditional universe, it has
to be the conditional density.
Now let's try the second expression.
This is the probability that the Y falls
into the little interval.
So that's the density of Y times delta.
And then here we have an object which is the
conditional probability X in a universe where the value of Y
is given to us.
Now this relation is sort of approximate.
This is true for very small delta in the limit.
But we can cancel the deltas from both sides, and we're
left with a formula that links together PMFs and PDFs.
Now this may look terribly confusing because there's both
p's and f's involved.
But the logic should be clear.
If a random variable is discrete, it's
described by PMF.
So here we're talking about the PMF of X in some
particular universe.
X is discrete, so it has a PMF.
Similarly here.
Y is continuous so it's described by a PDF.
And even in the conditional universe where I tell you the
value of X, Y is still a continuous random variable, so
it's been described by a PDF.
So this is the basic relation that links
together PMF and PDFs.
In this mixed the world.
And now in this inequality, you can take this term and
send it to the new denominator to the other side.
And what you end up with is the formula
that we have up here.
And this is a formula that we can use to make inferences
about the discrete random variable X when we're told the
value of the continuous random variable Y. The probability
that X takes on a particular value has something
to do with the prior.
And other than that, it's proportional to this quantity,
the conditional of Y given X. So these are the quantities
that we plotted here.
Suppose that the x's are equally likely in your prior,
so we don't really care about that term.
It tells us that the posterior of X is proportional to that
particular density under the given x's.
So in this picture, if I were to get a particular y here, I
would say that x equals 1 has a probability that's
proportional to this quantity.
x equals 0 has a probability that's
proportional to this quantity.
So the ratio of these two quantities gives us the
relative odds of the different x's given the y
that we have observed.
So we're going to come back to this topic and redo plenty of
examples of these kinds towards the end of the class,
when we spend some time dedicated
to inference problems.
But already at this stage, we sort of have the basic skills
to deal with a lot of that.
And it's useful at this point to pull all
the formulas together.
So finally let's look at the last case that's remaining.
Here we have a continuous phenomenon that we're trying
to measure, but our measurements are discrete.
What's an example where this might happen?
So you have some device that emits light, and you drive it
with a current that has a certain intensity.
You don't know what that current is, and it's a
continuous random variable.
But the device emits light by sending
out individual photons.
And your measurement is some other device that counts how
many photons did you get in a single second.
So if we have devices that emit a very low intensity you
can actually start counting individual photons as they're
being observed.
So we have a discrete measurement, which is the
number of problems, and we have a continuous hidden
random variable that we're trying to estimate.
What do we do in this case?
Well we start again with a formula of this kind, and send
the p term to the denominator.
And that's the formula that we use there, except that the
roles of x's and y's are interchanged.
So since here we have Y being discrete, we should change all
the subscripts.
It would be p_Y f_X given y f_X, and P(Y given X).
So just change all those subscripts.
Because now what we're used to be continuous became discrete,
and vice versa.
Take that formula, send the other terms to the
denominator, and we have a formula for the density, or X,
given the particular measurements for Y that we
have obtained.
In some sense that's all there is in Bayesian inference.
It's using these very simple one line formulas.
But why are there people then who make their living solving
inference problems?
Well, the devil is in the details.
As we're going to discuss, there are some real world
issues of how exactly do you design your f's, how do you
model your system, then how do you do your calculations.
This might not be always easy.
For example, there's certain integrals or sums that have to
be evaluated, which may be hard to do and so on.
So this object is a lot of richer
than just these formulas.
On the other hand, at the conceptual level, that's the
basis for Bayesian inference, that these
are the basic concepts.
All right, so now let's change gear and move to the new
subject, which is the topic of finding the distribution of a
functional for a random variable.
We call those distributions derived distributions, because
we're given the distribution of X. We're interested in a
function of X. We want to derive the distribution of
that function based on the distribution
that we already know.
So it could be a function of just one random variable.
It could be a function of several random variables.
So one example that we are going to solve at some point,
let's say you have to run the variables X and Y. Somebody
tells you their distribution, for example, is a uniform of
the square.
For some reason, you're interested in the ratio of
these two random variables, and you want to find the
distribution of that ratio.
You can think of lots of cases where your random variable of
interest is created by taking some other unknown variables
and taking a function of them.
And so it's legitimate to care about the distribution of that
random variable.
A caveat, however.
There's an important case where you don't need to find
the distribution of that random variable.
And this is when you want to calculate the expectations.
If all you care about is the expected value of this
function of the random variables, you can work
directly with the distribution of the original random
variables without ever having to find the PDF of g.
So you don't do unnecessary work if it's not needed, but
if it's needed, or if you're asked to do it,
then you just do it.
So how do we find the distribution of the function?
As a warm-up, let's look at the discrete case.
Suppose that X is a discrete random variable and takes
certain values.
We have a function g that maps x's into y's.
And we want to find the probability mass function for
Y.
So for example, if I'm interested in finding the
probability that Y takes on this particular value, how
would they find it?
Well I ask, what are the different ways that these
particular y value can happen?
And the different ways that it can happen is either if x
takes this value, or if X takes that value.
So we identify this event in the y space with that event in
the x space.
These two events are identical.
X falls in this set if and only if Y falls in that set.
Therefore, the probability of Y falling in that set is the
probability of X falling in that set.
The probability of X falling in that set is just the sum of
the individual probabilities of the x's in this set.
So we just add the probabilities of the different
x's where the summation is taken over all x's that leads
to that particular value of y.
Very good.
So that's all there is in the discrete case.
It's a very nice and simple.
So let's transfer these methods to
the continuous case.
Suppose we are in the continuous case.
Suppose that X and Y now can take values anywhere.
And I try to use same methods and I ask, what is the
probability that Y is going to take this value?
At least if the diagram is this way, you would say this
is the same as the probability that X takes this value.
So I can find the probability of Y being this in terms of
the probability of X being that.
Is this useful?
In the continuous case, it's not.
Because in the continuous case, any single value has 0
probability.
So what you're going to get out of this argument is that
the probability Y takes this value is 0, is equal to the
probability that X takes that value which also 0.
That doesn't help us.
We want to do something more.
We want to actually find, perhaps, the density of Y, as
opposed to the probabilities of individual y's.
So to find the density of Y, you might argue as follows.
I'm looking at an interval for y, and I ask what's the
probability of falling in this interval.
And you go back and find the corresponding set of x's that
leads to those y's, and equate those two probabilities.
The probability of all of those y's collectively should
be equal to the probability of all of the x's that map into
that interval collectively.
And this way you can relate the two.
As far as the mechanics go, in many cases it's easier to not
to work with little intervals, but instead to work with
cumulative distribution functions that used to work
with sort of big intervals.
So you can instead do a different picture.
Look at this set of y's.
This is the set of y's that are smaller
than a certain value.
The probability of this set is given by the cumulative
distribution of the random variable Y.
Now this set of y's gets produced by some corresponding
set of x's.
Maybe these are the x's that map into y's in that set.
And then we argue as follows.
The probability that the Y falls in this interval is the
same as the probability that X falls in that interval.
So the event of Y falling here and the event of X falling
there are the same, so their probabilities must be equal.
And then I do the calculations here.
And I end up getting the cumulative distribution
function of Y. Once I have the cumulative, I can get the
density by just differentiating.
So this is the general cookbook procedure that we
will be using to calculate it derived distributions.
We're interested in a random variable Y, which is a
function of the x's.
We will aim at obtaining the cumulative distribution of Y.
Somehow, manage to calculate the probability of this event.
Once we get it, and what I mean by get it, I don't mean
getting it for a single value of little y.
You need to get this for all little y's.
So you need to get the function itself, the
cumulative distribution.
Once you get it in that form, then you can calculate the
derivative at any particular point.
And this is going to give you the density of Y.
So a simple two-step procedure.
The devil is in the details of how you carry the mechanics.
So let's do one first example.
Suppose that X is a uniform random variable, takes values
between 0 and 2.
We're interested in the random variable Y, which is the cube
of X. What kind of distribution
is it going to have?
Now first notice that Y takes values between 0 and 8.
So X is uniform, so all the x's are equally likely.
You might then say, well, in that case, all the y's should
be equally likely.
So Y might also have a uniform distribution.
Is this true?
We'll find out.
So let's start applying the cookbook procedure.
We want to find first the cumulative distribution of the
random variable Y, which by definition is the probability
that the random variable is less than or equal to a
certain number.
That's what we want to find.
What we have in our hands is the distribution of X. That's
what we need to work with.
So the first step that you need to do is to look at this
events and translate it, and write it in terms of the
random variable about which you know you have information.
So Y is X cubed, so this event is the same as that event.
So now we can forget about the y's.
It's just an exercise involving a single random
variable with a known distribution and we want to
calculate the probability of some event.
So we're looking at this event.
X cubed being less than or equal to Y. We massage that
expression so that's it involves X directly, so let's
take cubic roots of both sides of this inequality.
This event is the same as the event that X is less than or
equal to Y to the 1/3.
Now with a uniform distribution on [0,2], what is
that probability going to be?
It's the probability of being in the interval from 0 to y to
the 1/3, so it's going to be in the area under the uniform
going up to that point.
And what's the area under that uniform?
So here's x.
Here is the distribution of X. It goes up to 2.
The distribution of X is this one.
We want to go up to y to the 1/3.
So the probability for this event happening is this area.
And the area is equal to the base, which is y to the 1/3
times the height.
What is the height?
Well since the density must integrate to 1, the total area
under the curve has to be 1.
So the height here is 1/2, and that explains why we get the
1/2 factor down there.
So that's the formula for the cumulative distribution.
And then the rest is easy.
You just take derivatives.
You differentiate this expression with respect to y
1/2 times 1/3, and y drops by one power.
So you get y to 2/3 in the denominator.
So if you wish to plot this, it's 1/y to the 2/3.
So when y goes to 0, it sort of blows up and it
goes on this way.
Is this picture correct the way I've drawn it?
What's wrong with it?
[? AUDIENCE: Something. ?]
PROFESSOR: Yes.
y only takes values from 0 to 8.
This formula that I wrote here is only correct when the
preview picture applies.
I took my y to the 1/3 to be between 0 and 2.
So this formula here is only correct for y between 0 and 8.
And for that reason, the formula for the derivative is
also true only for a y between 0 and 8.
And any other values of why are impossible, so they get
zero density.
So to complete the picture here, the PDF of y has a
cut-off of 8, and it's also 0 everywhere else.
And one thing that we see is that the distribution of Y is
not uniform.
Certain y's are more likely than others, even though we
started with a uniform random
variable X. All right.
So we will keep doing examples of this kind, a sequence of
progressively more interesting or more complicated.
So that's going to continue in the next lecture.
You're going to see plenty of examples in your recitations
and tutorials and so on.
So let's do one that's pretty similar to the one that we
did, but it's going to add to just a small twist in how we
do the mechanics.
OK so you set your cruise control
when you start driving.
And you keep driving at the constants based at the
constant speed.
Where you set your cruise control is somewhere
between 30 and 60.
You're going to drive a distance of 200.
And so the time it's going to take for your trip is 200 over
the setting of your cruise control.
So it's 200/V.
Somebody gives you the distribution of V, and they
tell you not only it's between 30 and 60, it's roughly
equally likely to be anything between 30 and 60, so we have
a uniform distribution over that range.
So we have a distribution of V. We want to find the
distribution of the random variable T, which is the time
it takes till your trip ends.
So how are we going to proceed?
We'll use the exact same cookbook procedure.
We're going to start by finding the cumulative
distribution of T. What is this?
By definition, the cumulative distribution is the
probability that T is less than a certain number.
OK.
Now we don't know the distribution of T, so we
cannot to work with these event directly.
But we take that event and translate it into T-space.
So we replace the t's by what we know T to be in terms of V
or
the v's All right.
So we have the distribution of V. So now let's
calculate this quantity.
OK.
Let's massage this event and rewrite it as the probability
that V is larger or equal to 200/T.
So what is this going to be?
So let's say that 200/T is some number that
falls inside the range.
So that's going to be true if 200/T is bigger than 30, and
less than 60.
Which means that t is less than 30/200.
No, 200/30.
And bigger than 200/60.
So for t's inside that range, this number 200/t falls inside
that range.
This is the range of t's that are possible, given the
description of the problem the we have set up.
So for t's in that range, what is the probability that V is
bigger than this number?
So V being bigger than that number is the probability of
this event, so it's going to be the area under this curve.
So the area under that curve is the height of the curve,
which is 1/3 over 30 times the base.
How big is the base?
Well it's from that point to 60, so the base has a length
of 60 minus 200/t.
And this is a formula which is valid for those t's for which
this picture is correct.
And this picture is correct if 200/T happens to fall in this
interval, which is the same as T falling in that interval,
which are the t's that are possible.
So finally let's find the density of T, which is what
we're looking for.
We find this by taking the derivative in this expression
with respect to t.
We only get one term from here.
And this is going to be 200/30, 1 over t squared.
And this is the formula for the density for t's in the
allowed to range.
OK, so that's the end of the solution to this particular
problem as well.
I said that there was a little twist compared to
the previous one.
What was the twist?
Well the twist was that in the previous problem we dealt with
the X cubed function, which was monotonically increasing.
Here we dealt with the function that was
monotonically decreasing.
So when we had to find the probability that T is less
than something, that translated into an event that
V was bigger than something.
Your time is less than something if and only if your
velocity is bigger than something.
So for when you're dealing with the monotonically
decreasing function, at some point some inequalities will
have to get reversed.
Finally let's look at a very useful one.
Which is the case where we take a linear function of a
random variable.
So X is a random variable with given distribution, and we can
see there is a linear function.
So in this particular instance, we take a to be
equal to 2 and b equal to 5.
And let us first argue just by picture.
So X is a random variable that has a given distribution.
Let's say it's this weird shape here.
And x ranges from -1 to +2.
Let's do things one step at the time.
Let's first find the distribution of 2X.
Why do you think you know about 2X?
Well if x ranges from -1 to 2, then the random variable X is
going to range from -2 to +4.
So that's what the range is going to be.
Now dealing with the random variable 2X, as opposed to the
random variable X, in some sense it's just changing the
units in which we measure that random variable.
It's just changing the scale on which we
draw and plot things.
So if it's just a scale change, then intuition should
tell you that the random variable X should have a PDF
of the same shape, except that it's scaled out by a factor of
2, because our random variable of 2X now has a range that's
twice as large.
So we take the same PDF and scale it up by stretching the
x-axis by a factor of 2.
So what does scaling correspond to
in terms of a formula?
So the distribution of 2X as a function, let's say, a generic
argument z, is going to be the distribution of X, but scaled
by a factor of 2.
So taking a function and replacing its arguments by the
argument over 2, what it does is it stretches it
by a factor of 2.
You have probably been tortured ever since middle
school to figure out when need to stretch a function, whether
you need to put 2z or z/2.
And the one that actually does the stretching is to put the
z/2 in that place.
So that's what the stretching does.
Could that to be the full answer?
Well there's a catch.
If you stretch this function by a factor of 2, what happens
to the area under the function?
It's going to get doubled.
But the total probability must add up to 1, so we need to do
something else to make sure that the area under the curve
stays to 1.
So we need to take that function and scale it down by
this factor of 2.
So when you're dealing with a multiple of a random variable,
what happens to the PDF is you stretch it according to the
multiple, and then scale it down by the same number so
that you preserve the area under that curve.
So now we found the distribution of 2X.
How about the distribution of 2X + 5?
Well what does adding 5 to random variable do?
You're going to get essentially the same values
with the same probability, except that those values all
get shifted by 5.
So all that you need to do is to take this PDF here, and
shift it by 5 units.
So the range used to be from -2 to 4.
The new range is going to be from 3 to 9.
And that's the final answer.
This is the distribution of 2X + 5, starting with this
particular distribution of X.
Now shifting to the right by b, what
does it do to a function?
Shifting to the right to by a certain amount,
mathematically, it corresponds to putting -b in the argument
of the function.
So I'm taking the formula that I had here, which is the
scaling by a factor of a.
The scaling down to keep the total area equal to 1.
And then I need to introduce this extra
term to do the shifting.
So this is a plausible argument.
The proof by picture that this should be the right answer.
But just in order to keep our skills tuned and refined, let
us do this derivation in a more formal way using our
two-step cookbook procedure.
And I'm going to do it under the assumption that a is
positive, as in the example that's we just did.
So what's the two-step procedure?
We want to find the cumulative of Y, and after that we're
going to differentiate.
By definition the cumulative is the probability that the
random variable takes values less than a certain number.
And now we need to take this event and translate it, and
express it in terms of the original random variables.
So Y is, by definition, aX + b, so we're
looking at this event.
And now we want to express this event in a clean form
where X shows up in a straight way.
Let's say I'm going to massage this event and
write it in this form.
For this inequality to be true, x should be less than or
equal to (y minus b) divided by a.
OK, now what is this?
This is the cumulative distribution of X evaluated at
the particular point.
So we got a formula for the cumulative Y based on the
cumulative of X. What's the next step?
Next step is to take derivatives of both sides.
So the density of Y is going to be the derivative of this
expression with respect to y.
OK, so now here we need to use the chain rule.
It's going to be the derivative of the F function
with respect to its argument.
And then we need to take the derivative of the argument
with respect to y.
What is the derivative of the cumulative?
The derivative of the cumulative
is the density itself.
And we evaluate it at the point of interest.
And then the chain rule tells us that we need to take the
derivative of this with respect to y, and the
derivative of this with respect to y is 1/a.
And this gives us the formula which is consistent with what
I had written down here, for the case where a
is a positive number.
What if a was a negative number?
Could this formula be true?
Of course not.
Densities cannot be negative, right?
So that formula cannot be true.
Something needs to change.
What should change?
Where does this argument break down when a is negative?
So when I write this inequality in this form, I
divide by a.
But when you divide by a negative number, the direction
of an inequality is going to change.
So when a is negative, this inequality becomes larger than
or equal to.
And in that case, the expression that I have up
there would change when this is larger than here.
Instead of getting the cumulative, I would get 1
minus the cumulative of (y minus b) divided by a.
So this is the probability that X is bigger than this
particular number.
And now when you take the derivatives, there's going to
be a minus sign that shows up.
And that minus sign will end up being here.
And so we're taking the negative of a negative number,
and that basically is equivalent to taking the
absolute value of that number.
So all that happens when we have a negative a is that we
have to take the absolute value of the scaling factor
instead of the factor itself.
All right, so this general formula is quite useful for
dealing with linear functions of random variables.
And one nice application of it is to take the formula for a
normal random variable, consider a linear function of
a normal random variable, plug into this formula, and what
you will find is that Y also has a normal distribution.
So using this formula, now we can prove a statement that I
had made a couple of lectures ago, that a linear function of
a normal random variable is also linear.
That's how you would prove it.
I think this is it for today so.